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+Strong Convergence of Peaks Over a Threshold
+S. A. Padoan
+Department of Decision Sciences, Bocconi University, Italy
+and
+S. Rizzelli
+Department of Statistical Sciences, Catholic University, Italy
+January 6, 2023
+Abstract
+Extreme Value Theory plays an important role to provide approximation re-
+sults for the extremes of a sequence of independent random variable when their
+distribution is unknown. An important one is given by the Generalised Pareto dis-
+tribution Hγ(x) as an approximation of the distribution Ft(s(t)x) of the excesses
+over a threshold t, where s(t) is a suitable norming function. In this paper we
+study the rate of convergence of Ft(s(t)·) to Hγ in variational and Hellinger dis-
+tances and translate it into that regarding the Kullback-Leibler divergence between
+the respective densities. We discuss the utility of these results in the statistical field
+by showing that the derivation of consistency and rate of convergence of estimators
+of the tail index or tail probabilities can be obtained thorough an alternative and
+relatively simplified approach, if compared to usual asymptotic techniques.
+Keywords: Contraction Rate, Consistency, Exceedances, Extreme Quantile, Gener-
+alised Pareto, Tail Index.
+2020 Mathematics Subject Classification: Primary 60G70; secondary 62F12, 62G20
+1
+Introduction
+Extreme Value Theory (EVT) develops probabilistic models and methods for describ-
+ing the random behaviour of extreme observations that rarely occur. These theoretical
+foundations are very important for studying practical problems in environmental, cli-
+mate, insurance and financial fields (e.g., Embrechts et al., 2013; Dey and Yan, 2016),
+to name a few.
+In the univariate setting, the most popular approaches for statistical analysis are the
+so-called Block Maxima (BM) and Peaks Over Threshold (POT) (see e.g. B¨ucher and
+Zhou, 2021, for a review). Let X1, . . . , Xn be independent and identically distributed
+(i.i.d.) random variables according to a common distribution F. The first approach
+concerns the modelling of k sample maxima derived over blocks of a certain size m, i.e.
+Mm,i = max(X(i−1)m+1, . . . , Xim), i ∈ {1, . . . , k}. In this case, under some regularity
+conditions (e.g. de Haan and Ferreira, 2006, Ch. 1), the weak limit theory establishes
+that F m(amx+bm) converges pointwise to Gγ(x) as m → ∞, for every continuity point
+x of Gγ, where Gγ is the Generalised Extreme Value (GEV) distribution, am > 0 and
+bm are suitable norming constants for each m = 1, 2, . . . and γ ∈ R is the so-called
+tail index, which describes the tail heaviness of F (e.g. de Haan and Ferreira, 2006,
+Ch. 1). The second method concerns the modelling of k random variables out of the
+n available that exceed a high threshold t, or, equivalently, of k threshold excesses Yj,
+1
+arXiv:2301.02171v1  [math.PR]  5 Jan 2023
+
+j = 1, . . . , k, which are i.i.d. copies of Y = X −t|X > t. In this context, the Generalised
+Pareto (GP) distribution, say Hγ, appears as weak limit law of appropriately normalised
+high threshold exceedances, i.e. for all x > 0, Ft(s(t)x) converges pointwise to Hγ(x)
+as t → x∗, for all the continuity points x of Hγ(x), where Ft(x) = P(Y ≤ x) and
+s(t) > 0 is a suitable scaling function for any t ≤ x∗, with x∗ = inf(x : F(x) <
+∞). This result motivates the POT approach, which was introduced decades ago by
+the seminal paper Balkema and de Haan (1974). Since then, few other convergence
+results emerged.
+For instance, the uniform convergence of Ft(s(t) · ) to Hγ and the
+coresponding convergence rate have been derived by Pickands III (1975) and Raoult
+and Worms (2003), respectively. Similar results but in Wasserstein distance have been
+recently established by Bobbia et al. (2021). As for the GEV distribution, more results
+are available.
+In particular, there are sufficient conditions to ensure, in addition to
+weak convergence, that F m(am · +bm) converges to Gγ for example uniformly and
+in variational distance and the density of F m(am · +bm) converges pointwise, locally
+uniformly and uniformly to that of Gγ (e.g. Falk et al., 2010, Ch. 2; Resnick, 2007, Ch.
+2).
+The main contribution of this article is to provide new convergence results that can
+be useful in practical problems for the POT approach. Motivated by the utility in the
+statistical field to asses the asymptotic accuracy of estimation procedures, we study
+stronger forms of convergence than the pointwise one, as limt→x∗ D(Ft(s(t) · ), Hγ) = 0,
+where D( · ; · ) is either the variational distance, the Hellinger distance or the Kullback-
+Leibler divergence. In particular, we provide upper bounds for the rate of convergence
+to zero of D(Ft(s(t) · ); Hγ) in the case that D( · ; · ) is the variational and Hellinger dis-
+tance, and further translate them into bounds on Kullback-Leibler divergence between
+the densities of Ft(s(t)·) and Hγ, respectively.
+Estimators of the tail index γ (and other related quantities) are typically defined
+as functionals of the random variables (Y1, . . . , Yk), as for instance the popular Hill
+(Hill, 1975), Moment (Dekkers et al., 1989), Pickands (Pickands III, 1975), Maximum
+Likelihood (ML, Jenkinson, 1969), Generalised Probability Weighted Moment (GPWM,
+Hosking et al., 1985) estimators, to name a few. In real applications, the distribution
+F is typically unknown and so is F(s(t) · ). Although, for large t, Hγ provides a model
+approximation for Ft(s(t) · ), when one wants to derive asymptotic properties as the
+consistency and especially the rate of convergence of the tail index estimators (or other
+related quantities), still the fact that (after rescaling) the random variables (Y1, . . . , Yk)
+are actually distributed according to Ft(s(t) · ) needs to be taken into account, which
+makes asymptotic derivations quite burdensome. These are even more complicated if t
+is determined on the basis of the (k + 1)-th largest order statistic of the original sample
+X1, . . . , Xn, which is the most common situation in practical applications. In this case,
+the threshold is in fact random and, up to rescaling, Ft(s(t) · ) only gives a conditional
+model for the variables Yj given a fixed value t of the chosen statistic. Asymptotic
+properties for POT methods have been studied in the last fifty years, see for example
+Hall and Welsh (1984), Drees (1998), Dekkers and de Haan (1993) and the reference
+therein.
+Leveraging on our strong convergence results we can show that, for random sequences
+(such as sequences of estimators) convergence results in probability that hold under the
+limit model Hγ, are also valid for a rescaled sample of excesses over a large order statistic.
+Precisely, we show that the distribution of the latter, up to rescaling and reordering,
+is contiguous to that of an ordered i.i.d. sample from Hγ (e.g., van der Vaart, 2000,
+Ch. 6.2). As a by product of this result, one can derive the consistency and rate of
+convergence of a tail index estimator (or an estimator of a related quantity) by defining
+it as a functional of the random sequence (Z1, . . . , Zk) which is distributed according the
+2
+
+limit model Hγ, and, if density of Ft(s(t)·) satisfies some regularity conditions, then the
+same asymptotic results hold even when such estimator is defined through the sequence
+of excesses. This approach simplifies a lot the computations as asymptotic properties
+are easily derivable under the limit model.
+The article is organised as follows, Section 2 of the paper provides a brief summary
+of the probabilistic context on which our results are based. Section 3 provides our new
+results on strong convergence to a Pareto model. Section 4 explains in what applications
+concerning statistical estimation our results are useful. Section 5 provides the proofs of
+the main results.
+2
+Background
+Let X be a random variable with a distribution function F that is in the domain of
+attraction of the GEV distribution Gγ, shortly denoted as F ∈ D(Gγ). This means that
+there are norming constants am > 0 and bm ∈ R for m = 1, 2, . . . such that
+lim
+m→∞ F m(amx + bm) = exp
+�
+− (1 + γx)−1/γ�
+=: Gγ(x),
+(2.1)
+for all x ∈ R such that 1 + γx > 0, where γ ∈ R, and this is true if only if there is a
+scaling function s(t) > 0 with t < x∗ such that
+lim
+t→x∗ Ft(s(t)x) = 1 − (1 + γx)−1/γ =: Hγ(x),
+(2.2)
+e.g., de Haan and Ferreira (2006, Theorem 1.1.6). The densities of Hγ and Gγ are
+hγ(x) = (1 + γx)−(1/γ+1)
+and
+gγ(x) = Gγ(x)hγ(x),
+respectively. Let U(v) := F ←(1 − 1/v), for v ≥ 1, where F ← is the left-continuous
+inverse function of F and G←(exp(−1/x)) = (xγ − 1)/γ.
+Then, we recall that the
+first-order condition in formula (2.1) is equivalent to the limit result
+lim
+v→∞
+U(vx) − U(v)
+a(v)
+= xγ − 1
+γ
+,
+(2.3)
+for all x > 0, where a(v) > 0 is a suitable scaling function. In particular, we have that
+s(t) = a(1/(1 − F(t))), see de Haan and Ferreira (2006, Ch. 1) for possible selections of
+the function a.
+A stronger convergence form than that in formula (2.2) is the uniform one, i.e.
+sup
+x∈[0, x∗−t
+s(t) )
+|Ft(s(t)x) − Hγ(x)| → 0,
+t → x∗.
+To establish the speed at which Ft(s(t)x) converges uniformly to Hγ(x), Raoult and
+Worms (2003) relied on a specific formulation of the well-known second-order condi-
+tion. In its general form, the second order condition requires the existence of a posi-
+tive function a and a positive or negative function A, named rate function, such that
+limv→∞ |A(v)| = 0 and
+lim
+v→∞
+U(vx)−U(v)
+a(v)
+− xγ−1
+γ
+A(v)
+= D(x),
+x > 0,
+3
+
+where D is a non-null function which is not a multiple of (xγ − 1)/γ, see de Haan and
+Ferreira (2006, Definition 2.3.1). The rate function A is necessarily regularly varying at
+infinity with index ρ ≤ 0, named second-order parameter (de Haan and Ferreira, 2006,
+Theorem 2.3.3). In the sequel, we use the same specific form of second order condition
+of Raoult and Worms (2003) to obtain decay rates for stronger metrics than uniform
+distance between distribution functions.
+3
+Strong results for POT
+In this section, we discuss strong forms of convergence for the distribution of rescaled
+exceedances over a threshold. First, in Section 3.1, we discuss convergence to a GP
+distribution in variational and Hellinger distance, drawing a connection with known
+results for density convergence of normalized maxima. In Section 3.2 we quantify the
+speed of convergence in variational and Hellinger distance. Finally, in Section 3.3, we
+show how these can be used to also bound Kullback-Leibler divergences. Throughout,
+for a twice differentiable function W(x) on R, we denote with W ′(x) = (∂/∂x)W(x)
+and W ′′(x) = (∂2/∂x2)W(x) the first and second order derivatives, respectively.
+3.1
+Strong convergence under classical assumptions
+Let the distribution function F be twice differentiable. In the sequel, we denote f = F ′,
+gm = (F m(am · +bm))′ and ft = F ′
+t.
+Under the following classical von Mises-type
+conditions
+lim
+x→∞
+xf(x)
+1 − F(x) = 1
+γ ,
+γ > 0,
+lim
+x→x∗
+(x∗ − x)f(x)
+1 − F(x)
+= −1
+γ ,
+γ < 0,
+(3.1)
+lim
+x→x∗
+f(x)
+� x∗
+x (1 − F(v)dv)
+(1 − F(x))2
+= 0,
+γ = 0,
+we know that the first-order condition in formula (2.3) is satisfied and it holds that
+lim
+v→∞ va(v)f(a(v)x + U(v)) = (1 + γx)−1/γ−1
+(3.2)
+locally uniformly for (1 + γx) > 0. Since the equality gm(x) = F m−1(amx + bm)hm(x)
+holds true, with bm = U(m), am = a(m) and hm(x) = mamf(amx + bm), and since
+F m−1(amx+bm) converges to Gγ(x) locally uniformly as m → ∞, the convergence result
+in formula (3.2) thus implies that gm(x) converges to gγ(x) locally uniformly (Resnick,
+2007, Ch. 2.2).
+On the other hand, the density pertaining to Ft(s(t)x) is
+lt(x) := ft(s(t)x)s(t) = s(t)f(s(t)x + t)
+1 − F(t)
+and, setting v = 1/(1 − F(t)), we have a(v) = s(t) and v → ∞ as t → x∗. Therefore,
+a further implication of the convergence result in formula (3.2) is that lt(x) converges
+to hγ(x) locally uniformly for x > 0, if γ ≥ 0, or x ∈ (0, −1/γ), if γ < 0. In turn, by
+Scheffe’s lemma we have
+lim
+t→x∗ V (Pt, P) = 0,
+where
+V (Pt, P) = sup
+B∈B
+|Pt(B) − P(B)|
+4
+
+is the total variation distance between the probability measures
+Pt(B) := P
+�X − t
+s(t)
+∈ B
+����X > t
+�
+and P(B) := P(Z ∈ B),
+and where Z is a random variable with distribution Hγ and B is a set in the Borel
+σ-field of R, denoted by B. Let
+H 2(lt; hγ) :=
+� ��
+lt(x) −
+�
+hγ(x)
+�2
+dx
+be the square of the Hellinger distance. It is well know that the Hellinger and total
+variation distances are related as
+H 2(lt; hγ) ≤ 2V (Pt, P) ≤ 2H (lt; hγ),
+(3.3)
+see e.g. Ghosal and van der Vaart (2017, Appendix B). Therefore, the conditions in
+formula (3.1) ultimately entail that also the Hellinger distance between the density of
+rescaled peaks over a threshold lt and the GP density hγ converges to zero as t → x∗.
+In the next subsection we introduce a stronger assumption, allowing us to also quantify
+the speed of such convergence.
+3.2
+Convergence rates
+As in Raoult and Worms (2003) we rely on the following assumption, in order to derive
+the convergence rate for the variational and Hellinger distance.
+Condition 3.1. Assume that F is twice differentiable. Moreover, assume that there
+exists ρ ≤ 0 such that
+A(v) := vU′′(v)
+U ′(v) + 1 − γ
+defines a function of constant sign near infinity, whose absolute value |A(v)| is regularly
+varying as v → ∞ with index of variation ρ.
+When Condition 3.1 holds then the classical von-Mises conditions in formula (3.1)
+are also satisfied for the cases where γ is positive, negative or equal to zero, respec-
+tively. Furthermore, Condition 3.1 implies that an appropriate scaling function for the
+exceedances of a high threshold t < x∗, which complies with the equivalent first-order
+condition (2.2), is defined as
+s(t) = (1 − F(t))/f(t).
+With such a choice of the scaling function s, we establish the following results.
+Theorem 3.2. Assume Condition 3.1 is satisfied with γ > −1/2. Then, there exist
+constants ci > 0 with i = 1, 2, αj > 0 with j = 1, ..., 4, K > 0 and t0 < x∗ such that
+H 2(lt; hγ)
+K|A(v)|2 ≤ S(v)
+(3.4)
+for all t ≥ t0, where v = 1/(1 − F(t)) and
+S(v) :=
+�
+1 − |A(v)|α1 + 4 exp (c1|A(v)|α2) ,
+if γ ≥ 0
+1 − |A(v)|α3 + 4 exp (c2|A(v)|α4) ,
+if γ < 0
+.
+5
+
+Given the relationship between the total variation and Hellinger distances in (3.3),
+the following result is a direct consequence of Theorem (3.2).
+Corollary 3.3. Under the assumptions of Theorem 3.2, for all t ≥ t0
+V (Pt, P) ≤ |A(v)|
+�
+KS(v).
+Theorem 3.2 implies that the Hellinger and variational distances of the probability
+density and measure of rescaled exceedances from their GP distribution counterparts are
+bounded from above by C|A(v)|, for a positive constant C, as the threshold t approaches
+the end-point x∗. Since for a fixed x ∈ ∩t≥t0(0, x∗−t
+s(t) ) it holds that
+|Ft(s(t)x) − Hγ(x)| ≤ V (Pt, P)
+and since Raoult and Worms (2003, Theorem 2(i)) implies that |Ft(s(t)x)−Hγ(x)|/|A(v)|
+converges to a positive constant, there also exists c > 0 such that, for all large t, c|A(v)|
+is a lower bound for variational and Hellinger distances. Therefore, since
+c|A(v)| ≤ V (Pt, P) ≤ H (lt; hγ) ≤ C|A(v)|,
+the decay rate of variational and Hellinger distances is precisely |A(v)| as t → x∗.
+Differently from the result on uniform convergence in Raoult and Worms (2003),
+our results on convergence rates in the stronger total variation and Hellinger topologies
+are given for γ > −1/2. Although the bound in formula (3.4) remains mathematically
+valid also for tail indices below −1/2, the restriction γ > −1/2 is imposed to guarantee
+that constants α3, α4 in the definition of S(v) are positive, so that S(v) is positive and
+bounded as t approaches x∗. Note that such a behaviour of S is essential to deduce
+from the bound in formula (3.4) that the rate of convergence is |A(v)|.
+3.3
+Kullback-Leibler divergences
+A further implication of Theorem 3.2 concerns the speed of convergence to zero of the
+Kullback-Leibler divergence
+K (˜lt; hγ) :=
+�
+ln
+�
+˜lt(x)/hγ(x)
+�
+˜lt(x)dx,
+and the divergences of higher order p ≥ 2
+Dp(˜lt; hγ) :=
+� ���ln
+�
+˜lt(x)/hγ(x)
+����
+p ˜lt(x)dx,
+where ˜lt = (Ft(˜s(t) · ))′ and ˜s(t) is a scaling function possibly different from s(t), which
+ensures that the support of the conditional distribution Ft(˜s(t)x) is contained in that
+of the GP distribution Hγ when γ < 0, i.e. x ∈ R : x < (x∗ − t)/˜s(t) < −1/γ. We recall
+indeed that, when γ is negative, the end-point (x∗ − t)/s(t) of lt converges to −1/γ as
+t approaches x∗. Nevertheless, for t < x∗ it can be that (x∗ − t)/s(t) > −1/γ, entailing
+that K (lt; hγ) = Dp(lt; hγ) = ∞. The introduction of a more flexible scaling function
+˜s is thus meant to rule out this uninteresting situation. In order to exploit Theorem
+3.2 to give bounds on Kullback-Leibler and higher order divergences, we first introduce
+by the next two lemmas a uniform bound on density ratios and a Lipschitz continuity
+result.
+Lemma 3.4. Under the assumptions of Theorem 3.2, if ρ < 0 and γ ̸= 0, and if
+˜s(t)/s(t) → 1 as t → x∗, then there exist a t1 < x∗ and a constant M ∈ (0, ∞) such
+that
+sup
+t≥t1
+sup
+0<x< x∗−t
+˜s(t)
+˜lt(x)
+hγ(x) < M.
+6
+
+Lemma 3.5. Let γ > −1/2. Then, there exists ϵ > 0 and L > 0 such that
+H 2(hγ; hγ′(σ · )σ) < L2(|γ − γ′|2 + |1 − σ|2)
+whenever |γ − γ′|2 + |1 − σ|2 < ϵ2.
+Next, using the uniform bound on density ratio provided in Lemma 3.4 and the
+Lipschitz continuity property established in Lemma 3.5, we are able to translate the
+upper bounds on the squared Hellinger distance H 2(lt, hγ) into upper bounds on the
+Kullback-Leibler divergence K (˜lt; hγ) and higher order divergences Dp(˜lt; hγ).
+Corollary 3.6. Under the assumptions of Theorem 3.2 with in particular ρ < 0 and
+γ ̸= 0, if there also exists B > 0 such that, for all large t < x∗,
+|s(t)/˜s(t) − 1| ≤ B|A(v)|,
+then there exists a t2 < x∗ such that, for all t ≥ t2
+(a) K (˜lt; hγ) ≤ 2M(
+�
+KS(v) + BL)2|A(v)|2
+(b) Dp(˜lt; hγ) ≤ 2p!M(
+�
+KS(v) + BL)2|A(v)|2, with p ≥ 2.
+To extend the general results in Lemma 3.4 and Corollary 3.6 to the case of γ = 0
+seems to be technically over complicated.
+Nevertheless, there are specific examples
+where the properties listed in such lemmas are satisfied, such as the following one.
+Example 3.7. Let F(x) = exp(− exp(−x)), x ∈ R, be the Gumbel distribution function.
+In this case, Condition 3.1 is satisfied with γ = 0 and ρ = −1, so that Theorem 3.2
+applies to this example, and for an arbitrarily small ϵ > 0 we have
+lt(x)/h0(x) ≤ exp(exp(−t)) < 1 + ϵ
+for all x > 0 and suitably large t. Hence, the bounded density ratio property is satisfied
+and it is still possible to conclude that Dp(lt; h0)/|A(v)|2 and K (lt; h0)/|A(v)|2 can be
+bounded from above as in Corollary 3.6.
+4
+Implications
+From a statistical stand point, the results introduced in Sections 3 can be used to study
+consistency and rate of contraction of estimators of the true value for a quantity of
+interest relative to the distribution of threshold exceedances within a POT approach.
+First, in Section 4.1, we illustrate an application to a density estimation problem.
+Second, in Section 4.2, we discuss the problem of studying estimators’ asymptotic ac-
+curacy in more general terms. A by product of our theory in Section 3 is that the
+consistency of estimators of the GP distribution parameters or related quantities can
+be easily derived by means of a contiguity result (e.g. van der Vaart, 2000, Ch. 6),
+provided that appropriate regularity conditions are satisfied, avoiding complicated and
+long calculations, typically required for example by popular estimators of the tail index
+γ (Hall and Welsh, 1984; Drees, 1998; Dekkers and de Haan, 1993).
+4.1
+Density estimation
+Accurate density estimation for threshold excesses is a crucial problem for probabilis-
+tic foresting of extremes, and, in particular, for the construction of reliable predictive
+regions for future large observations. When a sample X1, . . . , Xn of i.i.d. random vari-
+ables, with a common distribution F, is available, a simple method to estimate the
+7
+
+density ft of (approximately) a small fraction k/n of exceedances, with k ∈ N, over a
+large quantile t = U(n/k), is as follows. Let X(n−k) < . . . < X(n) denote the k + 1
+largest order statistics of the sample. Then, for measurable functions Tk,i, i = 1, 2, let
+�γk = Tk,1(X(n−k), ..., X(n))
+be a generic estimator of the tail index γ and
+�sk = Tk,2(X(n−k), ..., X(n))
+be a generic estimator of the scaling function s(U(n/k)). Since under Condition 3.1 it
+holds that
+ft(x) ≈ hγ
+�
+x
+s(U(n/k))
+�
+1
+s(U(n/k)),
+then a plug-in estimator of ft(x) exploiting its GP approximation is given by
+�hk(x) := h�γk(x/�sk)(1/�sk).
+By means of Theorem 3.2 the accuracy of the above estimator can be assessed by
+quantifying its rate of contraction to the true density ft in Hellinger distance. This is
+formally stated by the next result.
+Proposition 4.1. Under the assumptions of Theorem 3.2 and assuming further that,
+for t = U(n/k) and k ≡ k(n), the following conditions are satisfied as n → ∞:
+(a) k → ∞ and k/n → 0,
+(b)
+√
+k|A(n/k)| → λ ∈ (0, ∞),
+(c) |�γk − γ| = Op(1/
+√
+k) and |�sk/s(U(n/k)) − 1| = Op(1/
+√
+k),
+it then holds that
+H (ft;�hk) = Op(1/
+√
+k).
+For some specific choices of the estimators �γk and �sk proposed in the literature
+on POT methods (e.g. de Haan and Ferreira, 2006, Ch.
+3–5), assumptions (a)–(b)
+of Proposition 4.1 have been used along with the second order condition to establish
+asymptotic normality of the sequence
+√
+k
+�
+�γk − γ,
+�sk
+s(U(n/k)) − 1
+�
+.
+Such estimators thus comply with assumption (c) of Proposition 4.1, whose statement
+allows to readily obtain the rate of contraction of �hk to ft in Hellinger distance. We
+provide next two examples.
+Example 4.2. Under the assumptions of Theorem 3.2 and conditions (a)–(b) of Propo-
+sition 4.1, there exists a sequence of ML estimators of γ and s(U(n/k)) given by
+(�γk, �sk) ∈ arg max
+(γ,σ)∈D
+k
+�
+i=1
+hγ
+�X(n−k+i) − X(n−k)
+σ
+� 1
+σ
+where D = (−1/2, ∞) × (0, ∞), satisfying condition (c) of Proposition 4.1, see Drees
+et al. (2004) and Zhou (2009).
+8
+
+Example 4.3. The GPWM estimators of γ and s(U(n/k)) are defined as
+�γk = 1 −
+� Pk
+2Qk
+− 1
+�−1
+,
+�sk = Pk
+� Pk
+2Qk
+− 1
+�−1
+,
+where
+Pk = 1
+k
+k−1
+�
+i=0
+�
+X(n−i) − X(n−k)
+�
+,
+Qk = 1
+k
+k−1
+�
+i=0
+i
+k
+�
+X(n−i) − X(n−k)
+�
+.
+Under the assumptions of Theorem 3.2 and conditions (a)–(b) of Proposition 4.1, and
+assuming further that γ < 1/2, such estimators satisfy condition (c) of Proposition 4.1,
+see e.g. Theorem 3.6.1 in de Haan and Ferreira (2006).
+4.2
+Estimation consistency
+Popular estimators of the tail index γ as for example the Hill, Moment, Pickands, ML,
+GPWM (Hill, 1975; Dekkers et al., 1989; Pickands III, 1975; Jenkinson, 1969; Hosking
+et al., 1985), or estimators of other related quantities, are typically defined as suitable
+functionals of peaks/excesses over a large order statistic X(n−k), defined though the k
+larger statistics in a sample as
+Yk := (X(n−k+1) − X(n−k), . . . , X(n) − X(n−k)).
+Informally speaking, the random variable X(n−k) plays the role of a high threshold t
+and the sequence (X(n−k+i) − X(n−k)) with i = 1, . . . , k (up to rescaling) is seen as
+approximately distributed according to Hγ.
+Let Z1, . . . , Zk be a sample of i.i.d.
+random variables with GP distribution Hγ
+and let Zk = (Z(1), . . . , Z(k)) be the corresponding order statistics. In this section we
+establish the important statistical result that the distribution of the suitably rescaled
+sequence Yk is contiguous to that of the sequence Zk. To this aim, we first recall the
+notion of contiguity, see van der Vaart (e.g., 2000, Ch. 6.2) for more details.
+Definition 4.4. Let Pk and Qk be two sequence of probability measures. Qk is said to
+be contiguous with respect to Pk, in symbols Pk ▷Qk, if for all measurable set sequences
+Ek for which Pk(Ek) = o(1) we also have Qk(Ek) = o(1).
+As in Proposition 4.1, in the sequel we assume k ≡ k(n) and k → ∞ as n → ∞.
+Proposition 4.5. Let Pk and Qk be the probability measures relative to the random
+sequences Zk and Yk/˜s(X(n−k)), respectively. Then, under the assumptions of Corollary
+3.6 and assumptions (a)–(b) of Proposition 4.1, we have that Pk ▷ Qk.
+In statistical problems where the aim is to estimate a functional of the limiting
+GP distribution, say θ := φ(Hγ), the contiguity result in Proposition 4.5 can be used
+to show that a suitable estimator Tk(Yk) of the parameter θ is consistent, or formally
+speaking D(Tk(Yk), θ) = op(1), for a suitable metric D of interest. The next result and
+the subsequent discussion illustrate this point.
+Corollary 4.6. Under the assumption of Proposition 4.5, if Tk is a scale invariant
+measurable function on (0, ∞)k and Tk(Zk) is consistent estimator of θ as n → ∞, then
+also Tk(Yk) is a consistent estimator of θ as n → ∞.
+In real applications the distribution F of the original sample (X1, . . . , Xn) is typically
+unknown and as a result also the distribution of Yk is unknown.
+For this reason,
+9
+
+proving consistency of an estimator of the form Tk(Yk) for the parameter θ can be quite
+burdensome, and this is especially true for the derivation of its rate of contraction. We
+recall that quantifying the speed of convergence, or contraction rate, of an estimator
+Tk(Yk) of a parameter θ concerns the derivation of a positive sequences ϵk such that
+ϵk ↓ 0 and D(Tk(Yk), θ) = Op(ϵk) as k → ∞, for a suitable metric D.
+On the contrary, to establish the consistency of an estimator of the form Tk(Zk)
+for estimating θ and its contraction rate is much easier, and these preliminary results
+can be readily extended to the more demanding estimator Tk(Yk) by our Corollary 4.6,
+therefore establishing its consistency and the associated speed of convergence.
+We conclude the section with the following remark. It should be noted that within
+the POT approach it is common to use estimators defined on the basis of scale invariant
+functionals Tk. This is the case for many estimators of the tail index γ as those afore-
+mentioned. Nevertheless, the result of Proposition 4.5 extends also to estimators which
+are not invariant to rescaling of the data, provided that the discrepancy D(Tk(Yk), θ)
+can be suitably decomposed into several terms that depends on Yk/˜s(X(n−k)) up to an
+op(1) reminder.
+5
+Proofs
+5.1
+Additional notation
+For y > 0, we denote T(y) = U(ey) and, for t < x∗, we define the functions
+pt(y) =
+� T(y+T −1(t))−t
+s(t)
+− eγy−1
+γ
+,
+γ ̸= 0
+T(y+T −1(t))−t
+s(t)
+− y,
+γ = 0
+,
+with s(t) = (1 − F(t))/f(t), and
+qt(y) =
+�
+1
+γ ln [1 + γe−γypt(y)] ,
+γ ̸= 0
+pt(y),
+γ = 0
+.
+Moreover, for x ∈ (0, x∗ − t), we let φt(x) = T −1(x + t) − T −1(t). Finally, for x ∈ R,
+γ ∈ R, ρ ≤ 0 and σ > 0, we set
+Iγ,ρ(x) =
+� x
+0
+eγs
+� s
+0
+eρzdzds
+and ψx,γ = νx/σ(γ)
+�
+1/σ, with
+νx(γ) =
+��
+hγ(x),
+1 + γx > 0
+0,
+otherwise
+.
+5.2
+Auxiliary results
+In this section we provide some results which are auxiliary to the proofs of the main ones,
+presented in Section 3. Throughout, for Lemmas 5.1–5.6, Condition 3.1 is implicitly
+assumed to hold true.
+Lemma 5.1. For every ε > 0 and every α > 0, if γ ≥ 0, or α ∈ (0, −1/γ), if γ < 0,
+there exist x1 < x∗ and κ1 > 0 such that, for all t ≥ x1 and y ∈ (0, −α ln |A(eT −1(t))|)
+(a) if γ ≥ 0, then
+eqt(y) ∈
+�
+e±κ1|A(eT −1(t))|e2εy�
+;
+10
+
+(b) if γ < 0, then
+eqt(y) ∈
+�
+e±κ1|A(eT −1(t))|e(γ−ε)y�
+.
+Proof. By Lemma 5 in Raoult and Worms (2003), for all ε > 0 there exists x0 such that
+for all t ∈ (x0, x∗) and y > 0,
+e−γx|pt(y)| ≤ (1 + ε)|A(eT −1(t))|Iγ,ρ(y)e(γ−ε)y.
+Moreover, for a positive constant ϑ1
+Iγ,ρ(y)e(γ−ε)y ≤
+�
+ϑ1e2εy,
+γ ≥ 0
+ϑ1e(γ−ε)y,
+γ < 0
+.
+Combining these two inequalities, we deduce that
+e−γy|pt(y)| ≤
+�
+(1 + ε)|A(eT −1(t))|ϑ1e2εy,
+γ ≥ 0
+(1 + ε)|A(eT −1(t))|ϑ1e(γ−ε)y,
+γ < 0
+.
+(5.1)
+As a consequence, if γ ≥ 0, for any α > 0 there exists a constant ϑ2 such that
+sup
+y∈(0,−α ln |A(eT −1(t))|)
+e−γy|pt(y)| ≤ ϑ2|A(eT −1(t))|1−2εα
+(5.2)
+while, if γ < 0, for any α ∈ (0, −1/γ) there exists a constant ϑ3 such that
+sup
+y∈(0,−α ln |A(eT −1(t))|)
+e−γy|pt(y)| ≤ ϑ3|A(eT −1(t))|1−(ε−γ)α.
+(5.3)
+Therefore, choosing ε sufficiently small, e−γy|pt(y)| converges to zero uniformly over the
+interval (0, −α ln |A(eT −1(t))|) as t → x∗.
+It now follows that, if y ∈ (0, −α ln |A(eT −1(t))|) and t > x1 for a sufficiently large
+value x1 < x∗, when γ ̸= 0 a first-order Taylor expansion of the logarithm at 1 yields
+|qt(y)| =
+����
+1
+γ
+γe−γypt(y)
+1 + ϑ(t, y)γe−γypt(y)
+����
+≤
+�
+ϑ4|A(eT −1(t))|e2εy,
+γ > 0
+ϑ5|A(eT −1(t))|e(γ−ε)y,
+γ < 0
+,
+where ϑ(t, y) ∈ (0, 1) and ϑ4, ϑ5 are positive constants, while when γ = 0 it holds that
+|qt(y)| = eγye−γy|pt(y)|
+≤ ϑ6|A(eT −1(t))|e2εy,
+where ϑ6 is a positive constant. The two results in the statement are a direct consequence
+of the last two inequalities.
+Lemma 5.2. For every ε > 0 and every α > 0, if γ ≥ 0, or α ∈ (0, −1/γ), if γ < 0,
+there exist x2 < x∗ and κ2 > 0 such that, for all t ≥ x2 and y ∈ (0, −α ln |A(eT −1(t))|)
+(a) if γ ≥ 0, then
+1 + q′
+t(y) ∈
+�
+e±κ2|A(eT −1(t))|e2εy�
+;
+11
+
+(b) if γ < 0, then
+1 + q′
+t(y) ∈
+��
+e±κ2|A(eT −1(t))|e(γ−ε)y�
+.
+Proof. If γ ̸= 0
+1 + q′
+t(y) =
+exp
+�� ey+T −1(t)
+eT −1(t)
+A(u)
+u du
+�
+1 + γe−γypt(y)
+,
+while if γ = 0
+1 + q′
+t(y) = exp
+�� ey+T −1(t)
+eT −1(t)
+A(u)
+u
+du
+�
+.
+Therefore, if y ∈ (0, −α ln |A(eT −1(t))|) and t > x2 for a sufficiently large value x2 < x∗,
+using the bounds in formulas (5.1)–(5.3) and choosing a suitably small ε we deduce
+1 + q′
+t(y) ≤
+exp
+�� ey+T −1(t)
+eT −1(t)
+A(u)
+u du
+�
+1 − 1(γ ̸= 0)|γ|e−γy|pt(y)|
+≤ exp
+�
+y|A(eT −1(t))|
+�
+×
+�
+�
+�
+1
+1−ω1|A(eT −1(t))|e2εy ,
+γ ≥ 0
+1
+1−ω2|A(eT −1(t))|e(γ−ε)y ,
+γ < 0
+≤
+�
+�
+�
+exp
+�
+ω3|A(eT −1(t))|e2εy�
+,
+γ ≥ 0
+exp
+�
+ω4|A(eT −1(t))|e(γ−ε)y�
+,
+γ < 0
+for positive constants ωi, i = 1, . . . , 4. Similarly,
+1 + q′
+t(y) ≥
+exp
+�� ey+T −1(t)
+eT −1(t)
+A(u)
+u du
+�
+1 + 1(γ ̸= 0)|γ|e−γy|pt(y)|
+≥ exp
+�
+−y|A(eT −1(t))|
+�
+×
+�
+�
+�
+1
+1+ω5|A(eT −1(t))|e2εy ,
+γ ≥ 0
+1
+1+ω6|A(eT −1(t))|e(γ−ε)y ,
+γ < 0
+≥
+�
+�
+�
+exp
+�
+−ω7|A(eT −1(t))|e2εy�
+,
+γ ≥ 0
+exp
+�
+−ω8|A(eT −1(t))|e(γ−ε)y�
+,
+γ < 0
+for positive constants ωi, i = 5, . . . , 8. The result now follows.
+Lemma 5.3. If γ > 0 and ρ < 0, there exists a regularly varying function R with
+negative index ϱ such that, defining the function
+η(t) := (1 + γt)f(t)
+1 − F(t)
+− 1,
+as v → ∞, η(U(v)) = O(R(v)).
+Proof. Let v0 > 0 satisfy U(v0) ̸= 0 and U ′(v0) ̸= 0. Then, for v > v0 it holds that
+η(U(v)) = 1 + γU(v)
+vU′(v)
+− 1
+= 1 + γU(v0)
+vU′(v)
++ γ
+� v
+v0
+U ′(r)
+vU′(v)dr − 1.
+12
+
+Moreover, by definition of A, we have the identity
+γ
+� v
+v0
+U ′(r)
+vU′(v)dr − 1 =
+� 1
+v0/v
+U ′(zv)
+U ′(v) dz − 1
+=
+� 1
+v0/v
+γzγ−1
+�
+exp
+�
+−
+� 1
+z
+A(vu)
+u
+du
+�
+− 1
+�
+dz −
+�v0
+v
+�γ
+.
+Therefore, denoting by R2(v) the first term on the right-hand side and setting
+R1(v) = 1 + γU(v0)
+vU′(v)
+−
+�v0
+v
+�γ
+,
+we have η(U(v)) = R1(v) + R2(v).
+On one hand, the function R1(v) is regularly
+varying of order −γ. On the other hand, for any β ∈ (0, 1), the function R2(v) can be
+decomposed as follows
+R2(v) =
+� v−(1−β)
+v0/v
++
+� 1
+v−(1−β) γzγ−1
+�
+exp
+�
+−
+� 1
+z
+A(vu)
+u
+du
+�
+− 1
+�
+dz
+=: R2,1(v) + R2,2(v).
+Assuming that A is ultimately positive and selecting v0 suitably large, we have
+|R2,1(v)| ≤
+� v−(1−β)
+v0/v
+γzγ−1
+�
+1 − exp
+�
+−A(vz)
+z
+��
+dz
+= O(v−γ(1−β))
+and
+|R2,2(v)| ≤
+� 1
+v−(1−β) γzγ−1 �
+1 − zA(vβ)�
+dz
+= O(v−γ(1−β) ∨ A(vβ)).
+Consequently, there exists a regularly varying function R of index ϱ = γ(β − 1) ∨ ρβ
+complying with the property in the statement as v → ∞.
+Similarly, if A is ultimately negative, choosing β such that β < 2γ and v0 suitably
+large, we have
+|R2,1(v)| ≤
+� v−(1−β)
+v0/v
+γzγ−1 �
+uA(v0) − 1
+�
+dz
+= O(v−(γ−β/2)(1−β))
+and
+|R2,2(v)| ≤
+� 1
+v−(1−β) γzγ−1 �
+zA(vβ) − 1
+�
+dz
+= O(v−(γ−β/2)(1−β) ∨ |A(vβ)|)
+as v → ∞. Hence, there exists a regularly varying function R of index ϱ = (β − 1)(γ −
+β/2)∨ρβ complying with the property in the statement. The proof is now complete.
+Lemma 5.4. If γ > 0 and ρ < 0, there exists x3 ∈ (0, ∞) and δ > 0 such that, for all
+x ≥ x3,
+f(x) = hγ(x)
+�
+1 + O({1 − Hγ(x)}δ)
+�
+.
+13
+
+Proof. Let R∗(t) := R(1/(1 − F(t))), where R is as in Lemma 5.3. Then R∗(t) is regu-
+larly varying of index ϱ/γ (Resnick, 2007, Proposition 0.8(iv)). In turn, by Karamata’s
+theorem (e.g, Resnick, 2007, Proposition 0.6(a)) we have that for a large t∗
+� ∞
+t∗
+|η(t)|
+1 + γtdt < ∞
+and thus, by Proposition 2.1.4 in Falk et al. (2010), we conclude that
+τ := lim
+t→∞
+1 − F(t)
+1 − Hγ(t) ∈ (0, ∞).
+(5.4)
+As a consequence, for any δ ∈ (0, −ϱ), as t → ∞
+R∗(t) ∼ R
+�
+1
+τ(1 − Hγ(t))
+�
+= O({1 − Hγ(t)}δ).
+The conclusion now follows by Proposition 2.1.5 in Falk et al. (2010).
+Lemma 5.5. If γ < 0 and ρ < 0, there exists a a regularly varying function ˜R with
+negative index ˜ϱ = (−1) ∨ (−ρ/γ) such that, defining the function
+˜η(y) := (1 − γy)f(x∗ − 1/y)
+[1 − F(x∗ − 1/y)]y2 − 1,
+as y → ∞, ˜η(y) = O( ˜R(y)).
+Proof. By definition,
+˜η (y) =
+f(x∗ − 1/y)
+[1 − F(x∗ − 1/y)]y2 − γ
+�
+f(x∗ − 1/y)
+y(1 − F(x∗ − 1/y)) + 1
+γ
+�
+=: ˜η1 (y) + ˜η2 (y) .
+On one hand, we have that, as y → ∞
+˜η1 (y) = O(1/y).
+On the other hand, for v > 1 we have the identity
+˜η2
+�
+1
+x∗ − U(v)
+�
+=
+� ∞
+1
+γzγ−1
+�
+1 − exp
+�� z
+1
+A(uv)
+u
+du
+��
+dz.
+Hence, if A is ultimately positive,
+˜η2
+�
+1
+x∗ − U(v)
+�
+≤ −γ
+� ∞
+1
+zγ−1(zA(v) − 1)dz
+= O(A(v))
+while, if A is ultimately negative,
+����˜η2
+�
+1
+x∗ − U(v)
+����� ≤ γA(v)
+� ∞
+1
+zγ−1 ln zdz
+= O(|A(v)|).
+As a result of the two above inequalities, as v → ∞
+˜η2(t) = O
+�����A
+�
+1
+1 − F(x∗ − 1/y)
+�����
+�
+,
+Therefore, by regular variation of 1/(1 − F(x∗ − 1/y)) with index −1/γ, ˜η2(y) is even-
+tually dominated by a regularly varing function of index −ρ/γ. The final result now
+follows.
+14
+
+Lemma 5.6. If γ < 0 and ρ < 0, there exist ˜δ > 0 such that, as y → ∞,
+f(x∗ − 1/y)
+y2
+= (1 − γy)1/γ−1 �
+1 + O({1 − H−γ(y)}
+˜δ)
+�
+Proof. The function ˜f(y) := f(x∗ − 1/y)y−2 is the density of the distribution function
+˜F(y) := F(x∗−1/y), which is in the domain of attraction of G˜γ, with ˜γ = −γ. Moreover,
+˜η(y) = (1 + ˜γy) ˜f(y)
+1 − ˜F(y)
+− 1.
+By Lemma 5.5 and regular variation of 1 − H˜γ with index −1/˜γ, we have
+˜η(y) = O({1 − H˜γ(y)}
+˜δ)
+for any ˜δ > 0 such that −˜δ/˜γ > ˜ϱ. Therefore, by Proposition 2.1.5 in Falk et al. (2010),
+as y → ∞ it holds that
+˜f(y) = h˜γ(y)[1 + O({1 − H˜γ(y)}
+˜δ)],
+which is the result.
+Lemma 5.7. Let ν′
+x(γ) = (∂/∂γ)νx(γ).
+(a) If ξ : R �→ (0, ∞), then it holds that
+�� ∞
+0
+[ν′x(ξ(x))]2 dx ≤ 1
+2
+�� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x)
+�x ln(1 + xξ(x))
+ξ(x)
+�2
+dx
++ 1
+2
+�� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx.
+(b) If instead ξ : R �→ (γ∗, 0), then
+�� −1/γ∗
+0
+[ν′x(ξ(x))]2 dx ≤ 1
+2
+�� −1/γ∗
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x4dx
++ 1
+2
+�� −1/γ∗
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx.
+Proof. Let ϕx(γ) := (∂/∂γ) ln(1 − Hγ(x)). Then, for any x > 0, if ξ(·) > 0, or x ∈
+(0, −1/γ∗), if ξ(·) ∈ (γ∗, 0) we have
+ν′
+x(ξ(x)) = 1
+2(1 + xξ(x))− 1
+2 −
+1
+2ξ(x) ϕx(ξ(x)) − 1
+2(1 + xξ(x))− 3
+2 −
+1
+2ξ(x) x.
+If ξ(·) > 0, by Minkowski inequality
+�� ∞
+0
+[ν′x(ξ(x))]2 dx ≤ 1
+2
+�� ∞
+0
+(1 + xξ(x))−1−
+1
+ξ(x) [ϕx(ξ(x))]2 dx
++ 1
+2
+�� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx.
+15
+
+The result at point (a) now follows from the above inequality and the fact that, by
+equations (B.5)-(B.6) in B¨ucher and Segers (2017),
+0 ≤ ϕx(ξ(x)) ≤ x ln(1 + xξ(x))
+ξ(x)(1 + xξ(x)).
+If ξ(·) ∈ (γ∗, 0), inequality (B.8) in B¨ucher and Segers (2017) implies that for any
+x ∈ (0, −1/γ∗)
+0 ≤ ϕx(ξ(x)) ≤
+x2
+1 + xξ(x).
+This inequality and an argument by Minkowsi inequality, analogous to the previous one,
+now lead to the result at point (b).
+Lemma 5.8. Set ψ′
+γ,x(σ) = (∂/∂σ)ψγ,x(σ).
+(a) If ς : R �→ (1 ± ϵ), with ϵ ∈ (0, 1), and if γ > 0
+�� ∞
+0
+�
+ψ′γ,x(ς(x))
+�2 dx ≤
+�1
+γ +
+�
+1
+2γ + 1
+� �1 + ϵ
+1 − ϵ
+�5/2
+.
+(b) If γ ∈ (−1/2, 0) and ς(x) ∈ (σ∗, 1), with σ∗ ∈ (0, 1), there is a constant ζ > 0 such
+that
+�� − σ∗
+γ
+0
+�
+ψ′γ,x(ς(x))
+�2 dx ≤
+1
+σ∗
+√−γζ
+� 1
+γ2 + 1
+�
+.
+If instead ς(x) > 1,
+�� − 1
+γ
+0
+�
+ψ′γ,x(ς(x))
+�2 dx ≤
+1
+√−γζ
+� 1
+γ2 + 1
+�
+.
+Proof. Note that for x such that 1 + γx/σ > 0
+ψ′
+γ,x(σ) = (1 + γx/σ)− 1
+2γ − 3
+2
+σ5/2
+x
+γ + (1 + γx/σ)− 1
+2γ − 3
+2
+σ3/2
+.
+Consequently, if ς : R �→ (1 ± ϵ) and γ > 0, by Minkowski inequality
+�� ∞
+0
+�
+ψ′γ,x(ς(x))
+�2 dx
+≤
+�� ∞
+0
+(1 + γx/ς(x))− 1
+γ −3
+ς5(x)
+�x
+γ
+�2
+dx +
+�� ∞
+0
+(1 + γx/ς(x))− 1
+γ −3
+ς3(x)
+dx
+≤ (1 + ϵ)
+3
+2
+(1 − ϵ)
+5
+2
+1
+γ + (1 + ϵ)
+1
+2
+(1 − ϵ)
+3
+2
+�
+1
+2γ + 1
+� 1
+2
+and the result at point (a) follows.
+16
+
+If instead, ς(·) ∈ (σ∗, 1), for some σ∗ ∈ (0, 1), and γ ∈ (−1/2, 0), there is a constant
+ζ > 0 such that
+�� − σ∗
+γ
+0
+�
+ψ′γ,x(ς(x))
+�2 dx
+≤
+�� − σ∗
+γ
+0
+(1 + γx/ς(x))− 1
+γ −3
+ς5(x)
+�x
+γ
+�2
+dx +
+�� − σ∗
+γ
+0
+(1 + γx/ς(x))− 1
+γ −3
+ς3(x)
+dx
+≤
+�� − σ∗
+γ
+0
+(1 + γx/σ∗)−1+ζ
+σ3∗
+1
+γ4 dx +
+�� − σ∗
+γ
+0
+(1 + γx/σ∗)−1+ζ
+σ3∗
+dx
+=
+�
+1
+ζσ2∗(−γ)5 +
+�
+1
+ζσ2∗(−γ).
+The first half of the statement at point (b) is now established. The second half of the
+statement can be proved analogously.
+5.3
+Proof of Theorem 3.2
+For every xt > 0, it holds that
+H 2(lt; hγ) =
+� xt
+0
++
+� ∞
+xt
+��
+ft(x) −
+�
+hγ(x/s(t))/s(t)
+�2
+dx
+≤
+� φt(xt)
+0
+e−y
+�
+1 −
+�
+eqt(y)(1 + q′
+t(y))
+�2
+dy
++
+��
+1 − Ft(xt) +
+�
+1 − Hγ(xt/s(t))
+�2
+=: I1(t) + I2(t).
+Let xt be such that the following equality holds
+φt(xt) = −α ln |A(eT −1(t))|,
+for a positive constant α to be specified later. Then, by Lemmas 5.1-5.2, for a suitably
+small ε > 0 there exist constants κ3, κ4 > 0 such that for all sufficiently large t
+I1(t) ≤
+�
+�
+�
+� −α ln |A(eT −1(t))|
+0
+κ3|A(eT −1(t))|2e(4ε−1)ydy,
+γ ≥ 0
+� −α ln |A(eT −1(t))|
+0
+κ4|A(eT −1(t))|2e(γ−ε−1)ydy,
+γ < 0
+≤
+�
+�
+�
+κ3|A(eT −1(t))|2 �
+1 − |A(eT −1(t))|α1
+�
+,
+γ ≥ 0
+κ4|A(eT −1(t))|2 �
+1 − |A(eT −1(t))|α3
+�
+,
+γ < 0
+,
+where α1 := α(1 − 4ε) and α3 := α(1 − 2(ε − γ)) are positive. Moreover, on one hand
+we have the identity
+1 − Ft(xt) = |A(eT −1(t))|α.
+On the other hand, for some constants κ5, κ6 > 0 we have the inequality
+1 − Hγ(xt/s(t)) = |A(eT −1(t))|α exp
+�
+−qt
+�
+−α ln |A(eT −1(t))|
+��
+≤
+�
+�
+�
+|A(eT −1(t))|α exp
+�
+κ5|A(eT −1(t))|1−2εα�
+,
+γ ≥ 0
+|A(eT −1(t))|α exp
+�
+κ6|A(eT −1(t))|1−(ε−γ)α�
+,
+γ < 0
+.
+17
+
+Consequently,
+I2(t) ≤
+�
+�
+�
+|A(eT −1(t))|α �
+1 + exp
+�
+κ5
+2 |A(eT −1(t))|1−2εα��
+,
+γ ≥ 0
+|A(eT −1(t))|α �
+1 + exp
+�
+κ6
+2 |A(eT −1(t))|1−(ε−γ)α��
+,
+γ < 0
+.
+Now, if γ ≥ 0, we can choose α > 2 and ε small enough, so that
+|A(eT −1(t))|α < |A(eT −1(t))|2
+and α2 := 1 − 2εα > 0. If instead γ ∈ (−1/2, 0), we can choose α slightly larger than
+2 and ε small enough, so that the inequality in the above display is still satisfied and
+α4 := 1−α(ε−γ) > 0. The conclusion then follows noting that T −1(t) = − ln(1−F(t))
+and, in turn,
+|A(eT −1(t))| = |A(v)|.
+5.4
+Proof of Lemma 3.4
+We analyse the cases where γ > 0 and γ < 0 separately.
+Case 1: γ > 0. In this case, ˜s(t) = s(t) = (1 − F(t))/f(t) and ˜lt = lt. By Lemma
+5.4, there are positive constants κ, δ and ϵ such that, for all large t and all x > 0
+lt(x)
+hγ(x) ≤ hγ(s(t)x + t)
+hγ(x)
+s(t)
+1 − F(t)
+�
+1 + κ {1 − Hγ(s(t)x + t)}δ�
+≤
+�
+1 + γx
+(1 + γt)/s(t) + γx
+�1+1/γ
+1 + ϵ
+(s(t))1/γ(1 − F(t)).
+Moreover, by Lemma 5.3 it holds that as t → ∞
+1 + γt
+s(t)
+= 1 + η(t) = 1 + o(1)
+and, in turn, (s(t))1/γ ∼ (1+γt)1/γ. These two facts, combined with the tail equivalence
+relation in formula (5.4), imply that for all sufficiently large t and all x > 0
+lt(x)
+hγ(x) ≤
+�
+1 + γx
+1 − ϵ + γx
+�1+1/γ
+1 + ϵ
+(1 − ϵ)τ
+≤
+�
+1
+1 − ϵ
+�1+1/γ
+1 + ϵ
+(1 − ϵ)τ .
+The result now follows.
+Case 2: γ < 0. In this case, for any x ∈ (0, (x∗ − t)/˜s(t))
+˜lt(x) = f
+�
+x∗ − 1
+y
+� 1
+y2
+y2˜s(t)
+1 − F(t)
+where
+y ≡ y(x, t) :=
+1
+˜s(t)
+�x∗ − t
+˜s(t)
+− x
+�−1
+Note that y is bounded from below by 1/(x∗ − t), which converges to ∞ as t → x∗.
+Thus, by Lemma 5.6 there are positive constants ˜δ, ϵ and ˜κ such that
+˜lt(x) ≤ (1 − γy)1/γ−1[1 + ˜κ{1 − H−γ(y)}
+˜δ] y2˜s(t)
+1 − F(t)
+≤
+�
+1 + γ ˜s(t)
+x∗ − t
+�
+−1
+γ
+�
+x
+�− 1
+γ −1 �
+(x∗ − t)
+�
+1 − ˜s(t)x
+x∗ − t
+�
+− γ
+� 1
+γ −1 ˜s(t)(1 + ϵ)
+1 − F(t) (x∗ − t)− 1
+γ −1.
+18
+
+By hypothesis, it holds that
+x∗ − t
+˜s(t)
+≤ −1
+γ ,
+thus
+�
+1 + γ ˜s(t)
+x∗ − t
+�
+−1
+γ
+�
+x
+�− 1
+γ −1
+≤ (1 + γx)−1/γ−1.
+Moreover, it holds that
+1 − ˜s(t)x
+x∗ − t > 0,
+thus
+�
+(x∗ − t)
+�
+1 − ˜s(t)x
+x∗ − t
+�
+− γ
+� 1
+γ −1
+≤ (−γ)
+1
+γ −1.
+Finally, for all large t,
+˜s(t)
+x∗ − t ≤ −(1 + ϵ)γ.
+Combining all the above inequalities we can now conclude that, for all large t and for
+any x ∈ (0, (x∗ − t)/˜s(t)),
+˜lt(x)
+hγ(x) ≤ (1 + ϵ)2(−γ)
+1
+γ (x∗ − t)− 1
+γ
+1 − F(t) .
+Now, setting t = U(v), we have that v → ∞ if and only if t → x∗ and, by Theorem
+2.3.6 in de Haan and Ferreira (2006), there is a constant ϖ > 0 such that for all large t
+(x∗ − t)− 1
+γ
+1 − F(t)
+≤ v[(1 + ϵ)ϖvγ]− 1
+γ = [(1 + ϵ)ϖ]− 1
+γ
+The result now follows.
+5.5
+Proof of Lemma 3.5
+Note that for any γ′ > −1/2 and σ > 0
+H (hγ; hγ′(σ · )σ) ≤
+��
+R
+[νx(γ) − νx(γ′)]2 dx +
+��
+R
+[ψγ,x(σ) − ψγ,x(1)]2 dx
+In what follows, we bound the two terms on the right-hand side for γ′ ∈ (γ ± ϵ) and
+σ ∈ (1 ± ϵ), for a suitably small ϵ > 0. We study the the cases where γ > 0, γ < 0 and
+γ = 0 separately.
+Case 1: γ > 0. An application of the mean-value theorem and Lemma 5.7(a) yields
+that, for a function ξ(x) ∈ (γ ∧ γ′, γ ∨ γ′),
+��
+R
+[νx(γ) − νx(γ′)]2 dx = |γ − γ′|
+�� ∞
+0
+[ν′x(ξ(x))]2 dx
+≤ |γ − γ′|
+2
+�� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x)
+�x ln(1 + xξ(x))
+ξ(x)
+�2
+dx
++ |γ − γ′|
+2
+�� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx.
+19
+
+On one hand, it holds that
+� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x)
+�x ln(1 + xξ(x))
+ξ(x)
+�2
+dx
+≤ 4
+� ∞
+0
+(1 + x(γ − ϵ))−1−
+1
+γ+ϵ
+�ln(1 + x(γ − ϵ))
+(γ − ϵ)2
+�2
+dx
+≤ 8(γ + ϵ)3
+(γ − ϵ)5 .
+On the other hand, it holds that
+� ∞
+0
+(1 + ξ(x))−3−
+1
+ξ(x) x2dx
+���
+� ∞
+0
+(1 + x(γ − ϵ))−1−
+1
+γ+ϵ
+1
+(γ − ϵ)2 dx
+≤ (γ + ϵ)
+(γ − ϵ)3 .
+While, an application of the mean-value theorem and Lemma 5.8(a) yields that, for a
+function ς(x) ∈ (1 ∧ σ, 1 ∨ σ),
+�
+R
+�
+ψγ′,x(σ) − ψγ,x(1)
+�2 dx =
+� ∞
+0
+�
+ψ′
+γ,x(ς(x))
+�2 dx
+≤
+� 1
+γ2 +
+�
+1
+2γ + 1
+�2 �1 + ϵ
+1 − ϵ
+�5
+.
+The result now follows.
+Case 2: γ < 0. Assume that γ < γ′, then an application of the mean-value theorem
+and Lemma 5.7(b) yields that, for a function ξ(x) ∈ (γ, γ′),
+�
+R
+�
+νx(γ) − νx(γ′)
+�2 dx = |γ − γ′|2
+� −1/γ
+0
+�
+ν′
+x(ξ(x))
+�2 dx + 1 − Hγ′(−1/γ)
+≤ |γ − γ′|2
+4
+�
+�
+�� −1/γ
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x4dx
++
+�� −1/γ
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx
+�
+�
+2
++ 1 − Hγ′(−1/γ).
+First, for a constant β satisfying 0 < β < 1/(ϵ − γ) − 2, we have that
+� −1/γ
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x4dx ≤ 1
+γ4
+� −1/γ
+0
+(1 + γx)−1+βdx
+≤
+1
+(−γ)5
+1
+β .
+Similarly,
+� −1/γ
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx ≤ 1
+γ2
+� −1/γ
+0
+(1 + γx)−1+βdx
+≤
+1
+(−γ)3
+1
+β .
+20
+
+Finally, if ϵ is small enough, 1 − Hγ′(−1/γ) ≤ (1 − γ′/γ)2. Thus, we can conclude that
+�
+R
+�
+νx(γ) − νx(γ′)
+�2 dx ≤ |γ − γ′|2 1 + 1/2β
+(−γ)5 .
+A similar reasoning when γ > γ′ yields that
+�
+R
+�
+νx(γ) − νx(γ′)
+�2 dx ≤ |γ − γ′|2 1 + 2/β
+(−γ′)5
+≤ |γ − γ′|2 1 + 2/β
+(−γ − ϵ)5 .
+Next, assuming that σ < 1, an application of the mean-value theorem and the first
+half of Lemma 5.8(b) yields that for a function ς(x) ∈ (1 − ϵ, 1) and a constant ζ > 0
+�
+R
+[ψγ,x(σ) − ψγ,x(1)]2 dx = (1 − σ)2
+� −σ/γ
+0
+�
+ψ′
+γ,x(ς(x))
+�2 dx + (1 − σ)−1/γ
+≤ (1 − σ)2
+�
+1
+ζ(1 − ϵ)2(−γ)
+� 1
+γ2 + 1
+�2
++ 1
+�
+.
+While, if σ > 1, for a function ς(x) ∈ (1, 1 + ϵ)
+�
+R
+[ψγ,x(σ) − ψγ,x(1)]2 dx = (1 − σ)2
+� −1/γ
+0
+�
+ψ′
+γ,x(ς(x))
+�2 dx + (1 − 1/σ)−1/γ
+≤ (1 − σ)2
+�
+1
+ζ(−γ)
+� 1
+γ2 + 1
+�2
++ 1
+�
+.
+The result now follows.
+Case 3: γ = 0. Assume that γ′ > 0, then an application of the mean-value theorem
+and Lemma 5.7(a) yields that, for a function ξ(x) ∈ (0, γ′),
+�
+R
+�
+νx(γ) − νx(γ′)
+�2 dx = |γ − γ′|2
+� ∞
+0
+�
+ν′
+x(ξ(x))
+�2 dx
+≤ |γ − γ′|2
+4
+�
+�
+�� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x)
+�x ln(1 + xξ(x))
+ξ(x)
+�2
+dx
++
+�� −∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx
+�
+�
+2
+.
+On one hand, we have
+� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x)
+�x ln(1 + xξ(x))
+ξ(x)
+�2
+dx ≤
+� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x4dx
+≤
+� ∞
+0
+(1 + xγ′)−3− 1
+γ′ x4dx +
+� ∞
+0
+e−xx4dx
+≤ 36 + Γ(5).
+On the other hand, we have
+� ∞
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx ≤
+� ∞
+0
+(1 + xγ′)−3− 1
+γ′ x2dx +
+� ∞
+0
+e−xx2dx
+≤ 3 + Γ(3).
+21
+
+Assume next that γ′ < 0, then an application of the mean-value theorem and Lemma
+5.7(b) yields that, for a function ξ(x) ∈ (−ϵ, 0),
+�
+R
+�
+νx(γ) − νx(γ′)
+�2 dx = |γ − γ′|2
+� −1/γ′
+0
+�
+ν′
+x(ξ(x))
+�2 dx + e1/γ′
+≤ |γ − γ′|2
+4
+�
+�
+�� −1/γ′
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x4dx
++
+�� −1/γ′
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx
+�
+�
+2
++ e1/γ′.
+On one hand, for ϵ sufficiently small we have
+� −1/γ′
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x4dx ≤
+� −1/γ′
+0
+(1 + xγ′)−3− 1
+γ′ x4dx +
+� −1/γ′
+0
+e−xx4dx
+≤ 13
+2 Γ(5)
+and
+� −1/γ′
+0
+(1 + xξ(x))−3−
+1
+ξ(x) x2dx ≤
+� −1/γ′
+0
+(1 + xγ′)−3− 1
+γ′ x4dx +
+� −1/γ′
+0
+e−xx2dx
+≤ 3
+2Γ(3).
+On the other hand, for ϵ sufficiently small we have e1/γ′ ≤ |γ′ − γ|2.
+Finally, some algebraic manipulations yield
+�
+R
+[ψγ,x(σ) − ψγ,x(1)]2 dx =
+� ∞
+0
+��
+e−x/σ 1
+σ −
+√
+e−x
+�2
+dx
+≤ (1 − σ)2
+(1 − ϵ)2
+�
+1 + 1
+2
+�1 + ϵ
+1 − ϵ
+�3/2�2
+.
+The proof is now complete.
+5.6
+Proof of Corollary 3.6
+By Lemma 8.2 in Ghosal et al. (2000)
+K (˜lt; hγ) ≤ 2
+�
+�
+sup
+0<x< x∗−t
+˜s(t)
+˜lt(x)
+hγ(x)
+�
+� H 2(˜lt; hγ).
+Moreover, by Lemma B.3 in Ghosal and van der Vaart (2017), for p ≥ 2
+Dp(˜lt; hγ) ≤ 2p!
+�
+�
+sup
+0<x< x∗−t
+˜s(t)
+˜lt(x)
+hγ(x)
+�
+� H 2(˜lt; hγ).
+Furthermore, by triangular inequality and Lemma 3.5, for all large t
+H (˜lt; hγ) = H
+�
+lt; hγ
+�
+· s(t)
+˜s(t)
+� s(t)
+˜s(t)
+�
+≤ H (lt; hγ) + H
+�
+hγ; hγ
+�
+· s(t)
+˜s(t)
+� s(t)
+˜s(t)
+�
+≤ H (lt; hγ) + L|s(t)/˜s(t) − 1|
+≤ H (lt; hγ) + LB|A(v)|.
+22
+
+The conclusion now follows by combining the above inequalities and applying Theorem
+3.2 and Lemma 3.4.
+5.7
+Proof of Proposition 4.1
+By invariance of Hellinger distance under rescaling and triangle inequality
+H (ft;�hk) = H
+�
+lt; h�γk
+�
+· s(t)
+�sk(t)
+� s(t)
+�sk(t)
+�
+≤ H (lt; hγ) + H
+�
+hγ; h�γk
+�
+· s(t)
+�sk(t)
+� s(t)
+�sk(t)
+�
+.
+On one hand, by Theorem 3.2 and assumption (b), as n → ∞
+H (lt; hγ) = O(|A(n/k)|) = O(1/
+√
+k).
+Moreover, by Lemma 3.5 and assumption (c), as n → ∞
+H
+�
+hγ; h�γk
+�
+· s(t)
+�sk(t)
+� s(t)
+�sk(t)
+�
+= Op
+�
+�
+�
+|γ − �γk|2 +
+����1 − s(t)
+�sk(t)
+����
+2
+�
+�
+= Op(1/
+√
+k).
+The result now follows.
+5.8
+Proof of Proposition 4.5
+Let Qk denote the probability measure relative to the random sequence
+(Yk/˜s(X(n−k)), X(n−k)).
+Let Zk be the order statistics of an iid sample from Hγ, independent from X1, X2 . . .,
+and denote by Pk the probability measure relative to the random sequence (Zk, X(n−k)).
+In what follows, we prove that Pk ▷ Qk, which implies the result in the statement.
+We start by recalling that, as n → ∞,
+1/(1 − F(X(n−k)))
+n/k
+= 1 + op(1),
+see e.g. Lemma 2.2.3 in de Haan and Ferreira (2006). Hence, defining the set Bk :=
+(U((1 ± ϵ))n/k), for a small ϵ > 0, we have that for any measurable set sequence Ek
+Pk(Ek) = Pk(Ek|X(n−k) ∈ Bk)(1 + o(1)) + o(1)
+and
+Qk(Ek) = Pk(Ek|X(n−k) ∈ Bk)(1 + o(1)) + o(1)
+as n → ∞. Therefore, it suffices to prove that
+Pk( · |X(n−k) ∈ Bk) ▷ Qk( · |X(n−k) ∈ Bk).
+To do it, we denote by πk and χk the (Lebesgue) densities pertaining to the two condi-
+tional probability measures in the formula above and prove that
+lim sup
+n→∞ K (χk; πk) < ∞.
+(5.5)
+23
+
+Clearly, it holds that for almost every (y, t) ∈ Rk+2
+χk(y, t) = fYk/˜s(X(n−k))(y|X(n−k) = t)
+fX(n−k)(t)1(t ∈ Bk)
+P(X(n−k) ∈ Bk)
+,
+where fYk/˜s(X(n−k))(y|X(n−k) = t) and fX(n−k)(t) are the conditional density of Yk/˜s(X(n−k))
+given X(n−k) = t and the marginal density of X(n−k), respectively. Moreover,
+πk(y, t) = hZk(y)
+fX(n−k)(t)1(t ∈ Bk)
+P(X(n−k) ∈ Bk)
+,
+where hZk(y) is the density of Zk. As a consequence,
+K (χk; πk) =
+�
+Bk
+K (fYk/˜s(X(n−k))( · |X(n−k) = t); hZk)
+fX(n−k)(t)
+P(X(n−k) ∈ Bk)dt.
+By Lemma B.11 in Ghosal and van der Vaart (2017) and Lemma 3.4.1 in de Haan and
+Ferreira (2006)
+K (fYk/˜s(X(n−k))( · |X(n−k) = t); hZk) ≤ kK (˜lt; hγ).
+Moreover, by Corollary 3.6, there is a constant Λ > 0 such that for all large n
+sup
+t∈Bk
+K (˜lt; hγ) ≤ Λ
+���A
+�
+(1 − ϵ)n
+k
+����
+2
+≤ Λ(1 − ϵ)ρ(1 + ϵ)
+���A
+�n
+k
+����
+2
+.
+Combining the above inequalities we obtain that
+K (χk; πk) ≤ Λ(1 − ϵ)ρ(1 + ϵ)k
+���A
+�n
+k
+����
+2
+→ Λ(1 − ϵ)ρ(1 + ϵ)λ2
+as n → ∞, where the convergence result in the second line follows from assumption (b).
+The result in formula (5.5) is now established and the proof is complete.
+Acknowledgements
+Simone Padoan is supported by the Bocconi Institute for Data Science and Analytics
+(BIDSA), Italy.
+References
+Balkema, A. A. and L. de Haan (1974). Residual life time at great age. The Annals of
+probability 2, 792–804.
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+25
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+Single-material MoS2 thermoelectric junction enabled by substrate 
+engineering 
+Mohammadali Razeghi1, Jean Spiece3, Oğuzhan Oğuz1, Doruk Pehlivanoğlu2, Yubin Huang3, Ali 
+Sheraz2, Phillip S. Dobson4, Jonathan M. R. Weaver4, Pascal Gehring3, T. Serkan Kasırga1,2* 
+1 Bilkent University UNAM – Institute of Materials Science and Nanotehcnology, Bilkent 06800 Ankara, 
+Turkey 
+2 Department of Physics, Bilkent University, Bilkent 06800 Ankara, Turkey 
+3 IMCN/NAPS, Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium 
+4 James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, U.K 
+*Corresponding Author: kasirga@unam.bilkent.edu.tr 
+Abstract 
+To realize a thermoelectric power generator, typically a junction between two materials with different 
+Seebeck coefficient needs to be fabricated. Such difference in Seebeck coefficients can be induced by 
+doping, which renders difficult when working with two-dimensional (2d) materials. Here, we employ 
+substrate effects to form a thermoelectric junction in ultra-thin few-layer MoS2 films. We investigated 
+the junctions with a combination of scanning photocurrent microscopy and scanning thermal 
+microscopy. This allows us to reveal that thermoelectric junctions form across the substrate-
+engineered parts. We attribute this to a gating effect induced by interfacial charges in combination 
+with alterations in the electron-phonon scattering mechanisms. This work demonstrates that 
+substrate engineering is a promising strategy to develop future compact thin-film thermoelectric 
+power generators.  
+Main Text 
+In ultra-thin materials with large surface-to-bulk ratio, interactions with the substrate can have strong 
+impact on the materials properties 1–6. It is therefore important to understand this so-called substrate-
+effect, especially in order to optimize the reliability of future devices based on two-dimensional (2d) 
+semiconducting materials.  As an example, the choice of substrate for mono- and few-layer MoS2 has 
+been shown to strongly affect its Raman modes and photoluminescence (PL)7, electronic8, and thermal 
+transport9 properties. In this work, we employ the substrate effect to enable completely new 
+functionalities in a 2d semiconductor device. To this end, we engineer the substrate that atomically 
+thin MoS2 is deposited on. Using a combination of scanning photocurrent microscopy (SPCM) along 
+with scanning thermal microscopy (SThM) we demonstrate that substrate engineering is a powerful 
+way to build a thermoelectric junction. 
+
+ 
+Figure 1 a. Schematic of a substrate-engineered device: a MoS2 flake is suspended over a circular hole 
+drilled in the substrate. Metal contacts are used for scanning photocurrent microscopy (SPCM), 
+scanning thermal gate microscopy (SThGM) and I-V measurements. The inset shows a magnification 
+of the area indicated by the dashed yellow square, where Seebeck coefficients of supported and 
+suspended parts are labelled with 𝑆1 and 𝑆2, respectively. b. Optical microscope image of a multi-
+layered device over circular holes with indium contacts, marked with grey overlays. Scale bar: 10 µm. 
+c. SPCM reflection map and the corresponding open-circuit photocurrent map acquired from the 
+yellow dashed rectangle in b with 532 nm laser. {𝐼𝑚𝑖𝑛, 𝐼𝑚𝑎𝑥} = {−0.5, 0.5} nA. d. Photocurrent map 
+from the red dashed rectangle region in c. Black circle is the position of the hole determined from the 
+reflection image. Right panel shows the photocurrent, 𝐼𝑃𝐶 vs bias taken from point 1 (red dots) and 
+point 2 (blue dots) over the suspended part of the crystal marked on the left panel. Lower graph is the 
+derived photoconductance, 𝐺𝑃𝐶 vs. bias.  
+In the following we predict that a thermoelectric junction with a Seebeck coefficient difference of tens 
+of µV/K can be fabricated when connecting regions of suspended MoS2 to supported regions. We 
+assume that the Seebeck coefficient 𝑆 in thermal equilibrium is composed of contributions from the 
+energy-dependent diffusion (𝑆𝑁), scattering (𝑆τ)  and the phonon-drag (𝑆𝑝𝑑), so that 𝑆 = 𝑆𝑁 + 𝑆𝜏 +
+𝑆𝑝𝑑 9,10. Here, 𝑆𝑁 and 𝑆τ terms can be written from the Mott relation assuming that MoS2 is in the 
+highly conductive state and electrons are the majority carriers: 
+𝑆τ = −
+𝜋2𝑘𝐵
+2𝑇
+3𝑒
+𝜕ln𝜏
+𝜕𝐸 | 𝐸=𝐸𝐹 and 𝑆𝑁 = ±
+𝑘𝐵
+𝑒 [
+𝐸𝐹−𝐸𝐶
+𝑘𝐵𝑇 −
+(𝑟+2)𝐹𝑟+1(𝜂)
+(𝑟+1)𝐹𝑟(𝜂) ] 
+where 𝑇 is the temperature, 𝑘𝐵 is the Boltzmann constant, 𝑒 is the electron’s charge, 𝜏 is the relaxation 
+time, 𝐸𝐹 is the Fermi energy, 𝐸𝐶 is the conduction band edge energy, 𝑟 is scattering parameter and 𝐸 
+is the energy. 𝐹𝑚(𝜂) is the m-th order Fermi integral11. In the 2d limit, 𝜏 is energy independent, thus 
+𝑆𝜏 is zero. 𝑆𝑝𝑑 term can be estimated from the theory of phonon-drag in semiconductors in the first 
+order as 𝑆𝑝𝑑 = −
+𝛽𝑣𝑝𝑙𝑝
+𝜇𝑇   where, 𝑣𝑝 and 𝑙𝑝 are the group velocity and the mean free path of a phonon, 
+𝛽 is a parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and 𝜇 is 
+the electron mobility, respectively10. Importantly, 𝑙𝑝 and 𝜇 are heavily affected by the presence of a 
+substrate12 which implies that the 𝑆𝑝𝑑 term gets strongly modified when the MoS2 flake is suspended. 
+
+pended
+ported
+Reflection Map
+0mV
+S1
+noelectric
+Photocurrent Map
+0mVIndeed, we find that for suspended MoS2 at room temperature 𝑆𝑝𝑑 ≈ −100 µV/K and for MoS2 on 
+SiO2 at room temperature 𝑆𝑝𝑑 ≈ −230 µV/K. Similarly, 𝑆𝑁 is heavily influenced by the presence or 
+absence of the substrate as electron density depends on the interfacial Coulomb impurities and short-
+ranged defects11–17. We estimate that that for MoS2, 𝑆𝑁 ranges from -400 µV/K to -200 µV/K for carrier 
+concentrations ranging from 1012 cm-2 (suspended few layer MoS2) to 3 x 1013 cm-2 (SiO2 supported 
+few layer MoS2).18–20 As a result, a substrate engineered thermoelectric junction with a Seebeck 
+coefficient difference of Δ𝑆 ≈ 70 µV/K can be formed along the MoS2 flake (see Figure 1a and 
+Supporting Information).  
+To test this hypothesis, we fabricated substrate-engineered MoS2 devices by mechanical exfoliation 
+and dry transfer21 of atomically thin MoS2 flakes on substrates (sapphire or oxidized silicon) with pre-
+patterned trenches/holes formed by focused ion beam (FIB). We contacted the flakes with Indium 
+needles22–24 which are suitable for achieving Ohmic contacts to MoS225,26 (gold-contacted device 
+measurements are shown in Supporting Information). A typical device is shown in Figure 1b. We then 
+used scanning photocurrent microscopy, to locally heat up the junction with a focused laser beam and 
+to measure the photothermoelectric current that is generated (see Methods for experimental details). 
+Figure 1c shows the greyscale reflection intensity map and the corresponding photocurrent 
+distribution over the device. For the few-layer suspended MoS2 devices we observe a bipolar 
+photoresponse at the junctions between the supported and the suspended part of the crystal. The 
+spatial distribution of the signal agrees well with the finite element analysis simulations, given in the 
+supporting information, and suggests the formation of a thermoelectric junction. When applying a 
+voltage bias 𝑉 to the junction, the photocurrent, 𝐼𝑃𝐶 changes linearly with bias, while the 
+photoconductance, 𝐺𝑃𝐶 =
+𝐼𝑃𝐶
+𝑉 −𝐼𝑃𝐶
+0
+𝑉
+  (𝐼𝑃𝐶
+𝑉 , 𝐼𝑃𝐶
+0 : photocurrent under 𝑉 and 0 mV bias, respectively) stays 
+constant (Figure 1d). Such bias-independent photoconductance is typically an indication for an 
+photothermoelectric nature of the observed signal22,24,27–29. Although we propose that the 
+photocurrent in substrate-engineered MoS2 devices is dominated by the photothermal effect 
+(PTE)30,31, other possible mechanisms have been reported that may lead to a photovoltaic response. 
+These include (1) strain related effects such as strain modulation of materials properties and flexo-
+photovoltaic effect13, and (2) substrate proximity related effects that forms a built-in electric field32. 
+Next, we present experimental evidence for a thermoelectric origin of the observed photocurrent. To 
+this end, we employed scanning thermal gate microscopy (SThGM), where a hot AFM tip heats up the 
+junction locally while the resulting voltage build-up on the devices is recorded (see Methods). Since 
+no laser-illumination of the sample is required in this method, it can be used to ultimately exclude 
+photovoltaic effects. Figure 2 compares SPCM and SThGM maps of the same holes. We observed the 
+same bipolar signals in the suspended regions with both experimental methods. Thanks to its sub-100 
+nm lateral resolution, SThGM further allows us to observe local variations of the thermovoltage in 
+supported MoS2 that can be attributed to charge puddles induced by local doping via the substrate33–
+35. We confirmed that the SThGM signal disappears when no power is dissipated in the probe heater, 
+which rules out parasitic effects induced by the laser used for AFM feedback. Furthermore, SThGM 
+allows us to estimate the magnitude of the local Seebeck coefficient variations. Using the probe-
+calibration data we obtain a value of Δ𝑆 = 72 ± 10  µV/K (See supporting information).  Despite the 
+uncertainties regarding the real sample temperature, the obtained Δ𝑆 value is very close to the 
+theoretically predicted value. 
+
+ 
+Figure 2 a. SPCM reflection map and b. photocurrent map of the device shown in the inset of panel a. 
+Scale bar: 10 µm. The yellow rectangle indicates the region that was investigated by SThGM in c (AFM 
+height map) and d (SThGM thermovoltage map). e. SPCM map of the same region excerpted from the 
+map given in b. Color scale is the same as in panel b. Scale bars in c, d and e: 3 µm. 
+To understand why suspending MoS2 alters its Seebeck coefficient, we first would like to discuss the 
+possibility of strain induced changes in the materials properties. MoS2, like graphene, is nominally 
+compressed when deposited on a substrate36–39. Upon suspending the crystals, the free-standing part 
+either adheres to the sidewalls of the hole and dimples or, bulges. As a result, strain might be present 
+in the free-standing part of the crystal. Strain can affect both the bandgap and the Seebeck coefficient 
+of MoS2. The indirect optical gap is modulated by -110 meV/%-strain for a trilayer MoS236,40. Ab initio 
+studies show a ~10% decrease in the Seebeck coefficient of monolayer MoS2 per 1% tensile strain 41. 
+To estimate the biaxial strain, we performed atomic force microscopy (AFM) height trace mapping on 
+the samples. Most samples, regardless of the geometry of the hole exhibit slight bulging of a few 
+nanometers. For the MoS2 flakes suspended on the circular holes in the device shown in Figure 3a, 
+the bulge height is 𝛿𝑡 ≈ 25  nm. Similar 𝛿𝑡 values were measured for other devices. The biaxial strain 
+can then be calculated using an uniformly loaded circular membrane model, and is as low as 0.0025% 
+42. Such a small strain on MoS2 is not sufficient to induce a significant change in bandgap or Seebeck 
+coefficient 43–45.  
+Next, we consider the substrate induced changes on the material properties. The presence or the 
+absence of the substrate can cause enhanced or diminished optical absorption due to the screening 
+effects, Fermi level pinning46 and charges donated by the substrate7,47. More significantly, the doping 
+effect due to the trapped charges at the interface with the substrate can locally gate the MoS2 and 
+modify the number of charge carriers48 and thus its Seebeck coefficient. To investigate the 
+electrostatic impact of the substrate on the MoS2 membrane, we investigated the surface potential 
+difference (SPD) on devices using Kelvin Probe Force Microscopy (KPFM). SPD can provide an insight 
+on the band bending of the MoS2 due to the substrate effects49. Figure 3b-d shows the AFM height 
+trace map and the uncalibrated SPD map of the sample. SPD across the supported and suspended part 
+of the flake is on the order of 50 mV. This shift in the SPD value hints that there is a slight change in 
+the Fermi level of the suspended part with respect to the supported part of the crystal. The same type 
+of charge carriers is dominant on both sides of the junction formed by the suspended and supported 
+parts of the crystal. The band structure formed by such a junction in zero bias cannot be used in 
+separation of photoinduced carriers50, however, it can lead to the formation of a thermoelectric 
+junction11,51. This is in line with the SThGM measurements. 
+
+ 
+Figure 3 a. AFM height trace map of a device suspended over circular holes show a bulge of 𝛿𝑡 ≈ 25 
+nm. The line trace is overlayed on the map. Scale bar: 4 µm. b. AFM height trace map of the sample 
+shows the bulged and dimpled parts of the flake. Scale bar: 4 µm. c. KPFM map of the sample shows 
+the variation in the surface potential. Scale bar: 4 µm. d. Line traces taken along the numbered lines 
+in c. Direction of the arrows in c indicates the direction of the line plot.  
+In the remainder of the paper, we aim at controlling the electrostatics that are responsible for the 
+formation of a thermoelectric junction. Charge transport in MoS2 is dominated by electrons due to 
+unintentional doping52,53. Modulating the density and the type of free charge carriers can be done by 
+applying a gate voltage 𝑉𝑔 to the junction54. This significantly modifies the magnitude and the sign of 
+the Seebeck coefficient as demonstrated in previous studies16,30,31,55. The Mott relation56 can be used 
+to model the Seebeck coefficient as a function of 𝑉𝑔: 
+𝑆 =
+𝜋2𝑘𝐵
+2𝑇
+3𝑒
+1
+𝑅
+𝑑𝑅
+𝑑𝑉𝑔
+𝑑𝑉𝑔
+𝑑𝐸 | 𝐸=𝐸𝐹   eq.(1) 
+Here, 𝑇 is the temperature, 𝑘𝐵 is the Boltzmann constant, 𝑒 is the electron’s charge, 𝑅 is the device 
+resistance, 𝐸𝐹 is the Fermi energy and 𝐸 is the energy.  
+Since hole transport is limited due to substrate induced Fermi level pinning on SiO2 supported MoS2 
+field-effect devices,46 to observe the sign inversion of the Seebeck coefficient (see the Supporting 
+Information for measurements on device fabricated on SiO2 and Al2O3 coated SiO2) we followed an 
+alternative approach to emulate suspension: we fabricated heterostructure devices where the crystal 
+is partially supported by hexagonal boron nitride (h-BN). h-BN is commonly used to encapsulate two-
+dimensional materials thanks to its hydrophobic and atomically smooth surface. This leads to less 
+unintentional doping due to the interfacial charge trapping and reduced electron scattering7,57,58. A 
+~10 ML MoS2 is placed over a 10 nm thick h-BN crystal to form a double-junction device (see 
+supporting information for a single-junction device formed by a MoS2 flake which is partially placed 
+over a h-BN flake) and indium contacts are placed over the MoS2. The device is on 1 µm thick oxide 
+coated Si substrate where Si is used as the back-gate electrode. Figure 4a shows the optical 
+micrograph of the device and its schematic. The presence of h-BN modifies the SPD by 80 mV – a value 
+very similar to the values we find for suspended devices (see SI) – which is consistent with the relative 
+n-doping by the h-BN substrate32,57. We therefore attribute this difference to the Fermi level shift due 
+to the difference in interfacial charge doping by the different substrates.  
+
+ot
+0 1234μm 
+Figure 4 a. Optical micrograph of a Si back-gated MoS2 device partially placed over h-BN. Its cross-
+sectional schematic is shown in the lower panel. Scale bar: 10 µm. b. SPCM reflection map and the 
+photocurrent map of the device shown in a. 𝐼𝑚𝑎𝑥 = 3 nA and 𝐼𝑚𝑖𝑛 = −3 nA. Scale bar: 10 µm. c. 
+Current-Voltage graph versus 𝑉𝐺 from -40 to 40 V. Inset shows the resistance versus 𝑉𝐺. d. 𝐼𝑃𝐶 vs. 𝑉𝐺 
+recorded at the points marked in the SPCM map in b.  
+Figure 4b shows the SPCM map under zero gate voltage. We observe a bipolar photocurrent signal 
+from the junctions between h-BN and SiO2 supported MoS2. Raman mapping (see the Supporting 
+Information) reveals slight intensity decrease and a small shift of the A1𝑔 peak over the h-BN 
+supported part of the MoS2. This is consistent with the stiffening of the Raman mode due to the higher 
+degree of charged impurities in SiO2 as compared to h-BN7. By applying a gate voltage to the device, 
+its resistance can be tuned significantly as free charges are depleted (Figure 4c). Under large positive 
+gate voltages, the I-V characteristic becomes asymmetric. To investigate the dependence of the 
+photocurrent on carrier type and concentration, the laser is held at specific positions on the device as 
+marked in Figure 4d, and the gate is swept from positive to negative voltages with respect to the 
+ground terminal. For positive gate voltages, the magnitude of the photoresponse from both junctions, 
+between h-BN and SiO2 supported MoS2, (points 2 and 3) decrease. When a negative gate voltage is 
+applied, the magnitude of the photoresponse at both junctions increases by almost a factor of two at 
+𝑉𝐺 = −21.5 V. Once this maximum is reached, the amplitude of the photocurrent at both points 
+decreases and has the same value as the photocurrent generated over the MoS2 (point 4) at 𝑉𝐺 =
+ −34.5 V.  
+These observations can be qualitatively explained as follows: at a gate voltage of  𝑉𝐺 = −34.5 𝑉, the 
+majority charge carrier type in the h-BN supported part changes from electrons to holes. As a 
+consequence, the Seebeck coefficients of MoS2 resting on h-BN and SiO2, respectively, become similar, 
+which leads to ∆𝑆 ≈ 0, and curves 2,3 and 4 in Figure 4d cross. The photocurrent signal recorded near 
+the indium contacts (points 1 and 5) decreases non-monotonically with decreasing 𝑉𝐺 and reaches 
+
+SiO2
+Si
+Imax
+Iminzero at 𝑉𝐺 = −40 𝑉. At this voltage the Seebeck coefficient of MoS2 on SiO2 reaches that of Indium 
+(SIn = + 1.7 µV/K)59.  
+In conclusion we demonstrated that substrate engineering can be used to generate a thermoelectric 
+junction in atomically thin MoS2 devices. Similar strategies can be employed in other low dimensional 
+materials that exhibit large and tunable Seebeck coefficients. This might in particular be promising at 
+low temperature where effects like band-hybridization and Kondo scattering can produce a very 
+strong photothermoelectric effect9. 
+Author Contributions 
+T.S.K. designed and conceived the experiments, T.S.K. and P.G. prepared the manuscript. M.R. 
+fabricated devices, performed the experiment and analyzed the results. D.P. prepared the substrates, 
+performed simulations, and helped with the experiments. O.O. performed the AFM and KPFM 
+measurements and A.S. performed some of the earlier measurements. J.S., Y.H. and P.G. performed 
+the SThGM measurements and analyzed the results. P.S.D and J.M.R.W contributed discussions on the 
+implementation of VITA-DM-GLA-1 SThM probes. All authors discussed the results and reviewed the 
+final version of the manuscript. 
+Competing Interests 
+The Authors declare no Competing Financial or Non-Financial Interests. 
+Methods 
+SPCM setup is a commercially available setup from LST Scientific Instruments Ltd. which offers a 
+compact scanning head with easily interchangeable lasers. Two SR-830 Lock-in amplifiers are 
+employed, one for the reflection map and the other for the photocurrent/voltage measurements. In 
+the main text we reported the photocurrent (a measurement of the photovoltage is given in Figure 
+S2). The incident laser beam is chopped at a certain frequency and focused onto the sample through 
+a 40x objective. The electrical response is collected through gold probes pressed on the electrical 
+contacts of the devices and the signal is amplified by a lock-in amplifier set to the chopping frequency 
+of the laser beam. Although various wavelengths (406, 532, 633 nm) are employed for the 
+measurements, unless otherwise stated we used 532 nm in the experiments reported in the main text 
+(see Figure S3 for SPCM measurements with different wavelengths). All the excitation energies are 
+above the indirect bandgap of the few layer MoS2.  
+Scanning Thermal Microscopy measurements were performed with a Dimension Icon (Bruker) AFM 
+under ambient conditions. The probe used in the experiments is VITA-DM-GLA-1 made of a palladium 
+heater on a silicon nitride cantilever and tip. The radius is typically in the order of 25-40 nm. The heater 
+is part of a modified Wheatstone bridge and is driven by a combined 91 kHz AC and DC bias, as 
+reported elsewhere. The signal is detected via a SR830 lock-in amplifier and fed in the AFM controller. 
+This signal monitors the probe temperature and thus allows to locally map the thermal conductance 
+of the sample. In this work, the power supplied to the probe gives rise to a 45K excess temperature. 
+While the probe is scanning the sample, we measure the voltage drop across the device using a low 
+noise preamplifier (SR 560). This voltage is created by the local heating induced by the hot SThM tip. 
+It is then fed also to the AFM controller and recorded simultaneously. In this study, the thermovoltage 
+measurements were performed without modulating the heater power. We note that it is also possible 
+to generate similar maps by varying the heater temperature and detecting thermovoltage via lock-in 
+detection.  
+
+Data Availability  
+Source data available from the corresponding authors upon request. 
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+ 
+
+Supporting Information: Single-material MoS2 thermoelectric junction 
+enabled by substrate engineering 
+Mohammadali Razeghi1, Jean Spiece3, Oğuzhan Oğuz1, Doruk Pehlivanoğlu2, Yubin Huang3, Ali 
+Sheraz2, Phillip S. Dobson4, Jonathan M. R. Weaver4, Pascal Gehring3, T. Serkan Kasırga1,2* 
+1 Bilkent University UNAM – Institute of Materials Science and Nanotehcnology, Bilkent 06800 Ankara, 
+Turkey 
+2 Department of Physics, Bilkent University, Bilkent 06800 Ankara, Turkey 
+3 IMCN/NAPS, Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium 
+4 James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, U.K 
+*Corresponding Author: kasirga@unam.bilkent.edu.tr 
+ 
+1. Theoretical prediction of the substrate-effect induced Seebeck coefficient difference in MoS2 
+As discussed in the main text, we assume that the Seebeck coefficient S in thermal equilibrium is 
+composed of contributions from the energy-dependent diffusion (𝑆𝑁), scattering (𝑆τ)  and the phonon-
+drag (𝑆𝑝𝑑), so that 𝑆 = 𝑆𝑁 + 𝑆𝜏 + 𝑆𝑝𝑑. Here, 𝑆𝑁 and 𝑆τ terms can be written from the Mott relation 
+assuming that MoS2 is in the highly conductive state and electrons are the majority carriers: 
+𝑆τ = −
+𝜋2𝑘𝐵
+2𝑇
+3𝑒
+𝜕ln𝜏
+𝜕𝐸 | 𝐸=𝐸𝐹 and 𝑆𝑁 = ±
+𝑘𝐵
+𝑒 [
+𝐸𝐹−𝐸𝐶
+𝑘𝐵𝑇 −
+(𝑟+2)𝐹𝑟+1(𝜂)
+(𝑟+1)𝐹𝑟(𝜂) ] 
+As mentioned in the main text, 𝑆τ is zero as 𝜏 is energy independent in the 2d limit. 𝑆𝑁 term is 
+composed of constants related to material properties, scattering parameter 𝑟 and the Fermi integral 
+of the 𝑟-th order: 𝐹𝑟 = ∫
+ [
+𝑥𝑚
+𝑒𝑥−𝜂 + 1]𝑑𝑥
+∞
+0
+. The scattering parameters of 2d materials are listed in Table 
+1.1,2 Here, as discussed in detail in Ref. 1, 𝑟 = 0 adequately accounts for the acoustic phonon scattering 
+and small deviations of experimental data from the calculated values is due to the other scattering 
+mechanisms. As a result, at the room temperature 𝑆𝑁 for suspended MoS2 (1012 cm-2) is about -400 
+µV/K and for SiO2 supported MoS2 (1013 cm-2) is about -200 µV/K. 
+Table 1. Scattering parameters 𝒓 of 2d materials. 
+Scattering mechanism 
+𝒓 
+Charged Impurity Scattering 
+3/2 
+Acoustic Phonon Scattering 
+0 
+Intervalley Scattering 
+0 
+Strongly Screened Coulomb Scattering 
+-1/2 
+ 
+𝑆𝑝𝑑 term can be estimated from the theory of phonon-drag in semiconductors in the first order as 
+𝑆𝑝𝑑 = −
+𝛽𝑣𝑝𝑙𝑝
+𝜇𝑇   where, 𝑣𝑝 and 𝑙𝑝 are the group velocity and the mean free path of a phonon, 𝛽 is a 
+parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and 𝜇 is the 
+electron mobility, respectively.  As the dominant charge carriers are electrons, 𝑆𝑝𝑑 term has a negative 
+sign. We use the parameters given in Table 2. Based on the values given in the table we obtain 𝑆𝑝𝑑
+𝑆𝑖𝑂2 =
+ −230  µV/K and ���𝑝𝑑
+𝑆𝑢𝑠 = −100  µV/K.  
+
+The total 𝑆 = 𝑆𝑁 + 𝑆𝑝𝑑 for suspended and SiO2 supported parts can be calculated by adding both 
+contributions. 𝑆𝑆𝑢𝑠 = −500 µV/K and 𝑆𝑆𝑖𝑂2 = −430 µV/K. Of course, we consider this to be a rough 
+estimate as we ignore charged impurity scattering and strongly screeded Coulomb scattering. Also 
+there are certain errors associated with the measurement of the parameters used for the calculation 
+of the Seebeck coefficients. However, overall, this calculation shows that the substrate induced effect 
+must be present under right experimental conditions. 
+Table 2. Parameters used for 𝑺𝒑𝒅 calculation 
+Parameter 
+On SiO2 (Ref. 3) 
+Suspended (Ref. 3) 
+𝑣𝑝 
+7 105 cm/s 
+7 105 cm/s 
+𝑙𝑝 
+5 nm 
+20 nm 
+𝜇 
+5 cm2/V.s 
+50 cm2/V.s 
+ 
+2. SPCM map on a gold electrode substrate engineered MoS2 device 
+Throughout the study we used indium contacted devices thanks to their rapid fabrication. To compare 
+our indium device results, we fabricated gold contacted devices. Figure S1 shows the optical 
+microscope images and corresponding SPCM reflection and photocurrent maps. There is no qualitative 
+difference between the indium contacted devices and gold contacted device in the substrate-
+engineered photocurrent. Despite IV measurement is collected from 0.25 to -0.25 V its rectifying 
+behaviour can be observed. Power dependence of the photocurrent from the substrate engineered 
+junction is also comparable to the one reported in indium contacted devices. 
+ 
+Figure S1 a. Optical microscope micrograph of a gold contacted substrate engineered MoS2 device is 
+shown. Scale bar is 10 µm. b. SPCM reflection map and c. photocurrent map. d. IV curve shows signs 
+of rectifying nature of the contacts. e. Power dependence of the photocurrent at one of the side of 
+the junction is plotted in a log-log graph and the exponent is about 0.63.   
+
+b
+a 
+3. Scanning photovoltage microscopy, AFM and KPFM measurements on a parallel trench 
+device 
+Figure S2 shows an MoS2 device fabricated on trenches drilled on sapphire with different depths. We 
+performed SPVM, AFM and KPFM Measurements. First, AFM measurements show that the crystal is 
+stuck to the bottom of the 100 nm deep trench (Figure S2b). For the rest of the trenches the flake 
+bulges about 10 nm above the surface (Figure S2c). AFM height trace map also reveals a peculiar 
+wrinkle formation over the suspended part of the flake.  
+In this measurement we operated the scanning microscope at photovoltage mode. Figure S2d shows 
+the reflection map and the corresponding photovoltage map. The bipolar response is evident with 
+slightly lower positive signal in some of the trenches. This asymmetry can be explained by lower 
+heating of one side of the samples due to the scan direction. One important observation that agrees 
+well with the photothermoelectric photoresponse is that the 100 nm trench shows very small 
+photovoltage as compared to other trenches. 
+Figure S2e shows the KPFM and AFM profiles. The suspended part of the crystal has 60 meV lower 
+surface potential difference. This is consistent with other KPFM measurements. The lower panel shows 
+the variation in the height over the wrinkles. The workfunction is calculated with a calibrated tip and 
+it follows the wrinkles of the sample. However, the difference in the SPD is not due to the variations in 
+the height profile of the crystal. The change in the workfunciton is a good indication of the changes in 
+the electronic landscape of the device upon suspension. Small variations along the wrinkles are also 
+expected due to formation of varying stress regions along the crystal. 
+ 
+Figure S2 a. Optical microscope image of the device with trench depths labelled next to it. b. AFM 
+height trace map and c. line trace taken along the height trace. The bulge of the crystal over the 
+trenches is clear. d. Reflection and photovoltage maps obtained by operating the scanning microscope 
+in photovoltage mode. Scale bar is 5 µm. e. Left panel shows the workfunction and height taken over 
+
+Profile1
+103nm
+Onmthe red lines marked on the maps given in the right panel. The variation of the workfunction along the 
+trench is very small and correlated with the wrinkles of the crystal.  
+4. SPCM maps taken at different laser wavelengths and incidence polarization  
+We used three different wavelengths, 406, 532 and 633 nm, in our experiments all of them which are 
+at an energy larger than the band gap of MoS2. Figure S3 shows the SPCM results collected with 
+different laser wavelengths. Also, polarization dependence of the photocurrent measured at each end 
+of the trench as well as a point over the contact is given in Figure S3f. There is no polarization 
+dependence of the photocurrent. This shows that The effect is not due to built in polarization fields. 
+ 
+Figure S3 a. Optical microscope image of a two-terminal substrate-engineered MoS2 device with 
+different trench widths. Scale bar is 10 µm. b. SPCM reflection map of the region marked with yellow 
+rectangle in a. c, d and e show photocurrent map taken at different wavelengths. At each run laser 
+power is set to ~40 µW. The measured signal in all three measurements are very close and the overall 
+photocurrent features are the same. f. Incident polarization of the 633 nm laser is rotated and 𝐼𝑃𝐶 is 
+measured at three different points marked by colored arrows on d, black dots- near contact, red dots- 
+at the positive side and blue dots- at the negative side of the trench. There is no polarization 
+dependence of the measured photocurrent at the three points where photocurrent is measured. 
+5. Finite element simulation of a substrate modified thermoelectric junction 
+To understand how a substrate modified thermoelectric junction would behave depending on how the 
+contacts are configured, we performed finite element analysis simulations using COMSOL 
+Multiphysics. An irregularly shaped crystal is modelled over a substrate with a hole and voltage at a 
+floating terminal is measured with respect to different laser positions. The observed pattern agrees 
+with our measurements. Figure S4 shows the thermoelectric emf generated and temperature 
+distribution maps. 
+
+406 nm
+532 nm
+633 nm 
+Figure S4 a. An arbitrary crystal is modelled over a SiO2 substrate with a hole. The outline drawn over 
+the voltage distribution map shows the outline of the crystal and the outline of the hole. The red line 
+indicates the ground terminal and the blue line indicates the floating voltage terminal where the 
+photothermoelectric emf is measured from like in the experiments. b. For comparison, another 
+terminal is simulated as the floating terminal. c. Temperature distribution vs. laser position is shown. 
+As clear, the maximum temperature rise is achieved at the center of the hole.  
+6. KPFM on h-BN supported and suspended MoS2 
+ 
+Figure S5 a. Optical micrograph of an MoS2 crystal partially suspended over a trench and partially 
+supported on h-BN (outlined by blacked dashed lines). White dashed square shows the AFM region. b. 
+AFM height trace map and c. corresponding SPD map is given. d. SPD line traces from the colored lines 
+in c are plotted. The difference between the SPD in h-BN supported, suspended and SiO2 supported 
+parts are evident. Scale bars are 2 µm. 
+7. Gate dependent measurements 
+We performed gate dependent SPCM measurements both on suspended and h-BN supported MoS2 
+devices. In both cases, we used 1 µm SiO2 coated Si wafers. Si is used as the back gate in both device 
+configurations. We reported the h-BN supported junctions in the main text as devices over holes 
+showed significant change upon application of negative gate bias. Figure S5 shows the degradation of 
+the suspended device. After application of a few volts the device irreversibly shows a contrast change 
+
+hBN
+0.12
+0.10
+hBNSupported
+58mV
+Suspended
+:35mV
+Sio,Supported
+0.06
+0.04
+0.0
+0.5
+1.0
+1.5
+2.0
+lenght [um]starting from the edges of the hole. We fabricated a long trench with open ends to see if the trapped 
+air within the hole is causing the observed contrast change. However, same contrast change is 
+observed after applying negative gate voltages. We observe that the contrast change starts from near 
+the hole and expands from there. At the moment we are not fully aware of the reasons leading this 
+contrast change. We consider that the release of the adsorbed molecules on the surface of the 
+substrate under large negative gate voltages lead to such degradation.  
+ 
+Figure S6 a. Optical microscope micrograph of indium contacted MoS2 on SiO2/Si with stair-like holes 
+before and after application of gate voltages down to 𝑉𝐺 = −20 𝑉. Lower panel shows a clear contrast 
+change around the holes extending to the indium contacts. b. SPCM maps taken at 𝑉𝐺 = 0 𝑉 with 532 
+nm of 86 µW on sample: (i) before gating, (ii) after 𝑉𝐺 = −15 𝑉 scan and (iii) after the scan in (iii).   
+𝐼𝑚𝑎𝑥 = 6.5 nA and 𝐼𝑚𝑖𝑛 = −6.5 nA. Scan starts from top left corner to the bottom left corner with 
+progressing to the right in raster scan pattern. Scale bars are 5 µm. 
+To prevent the sample degradation problem under large negative gate voltages, we coated the 
+substrate surface with 5 nm thick Al2O3 using atomic layer deposition (ALD) method after milling the 
+holes with FIB. Then, the device is fabricated over the ALD coated surface. The device didn’t show any 
+sign of degradation and produced pronounced photoresponse. Measurements from the device is given 
+in Figure S6. Although the device exhibits the expected gate dependent response, as discussed in the 
+main text, there is no carrier inversion induced reduction in the photovoltage due to the Fermi level 
+pinning. 
+
+ii
+ili
+BeforeV
+After VG 
+Figure S7 a. Schematic of the device along with the optical image is shown. The sample is coated with 
+30 nm thick Al2O3 to passivate the SiO2 surface and to minimize the pinholes. Scale bar is 10 µm. b. 
+Photovoltage map collected in DC mode without the Lock-in amplifier and chopper. i is the reflection 
+map, and photovoltage maps at ii is the 𝑉𝐺 =-60 V, iii 𝑉𝐺 = 0 V, iv 𝑉𝐺 =60 V. Here, 𝑉𝑚𝑎𝑥 = 20 mV and 
+𝑉𝑚𝑖𝑛 = -20 mV. c. Photovoltage line trace taken along the dashed arrow given in b-ii. Large signal 
+corresponds to the more negative gate voltages. d. Photovoltage data collected from points indicated 
+on b-ii. This sample showed no Seebeck coefficient inversion due to possible Fermi level pinning 
+induced by the substrate as discussed in the main text. 
+H-BN supported devices performed better and showed no sign of such a contrast change. Figure S7 
+shows the reflection and the photocurrent maps reported in the main text and the photocurrent from 
+point 2 and 3 subtracted from point 4, marked on the photocurrent map. Both junctions of the h-BN 
+show almost identical response under gate voltage (point 3 data is multiplied by -1 for viewing 
+convenience). 
+ 
+
+ii
+MoS2
+B
+in
+In
+A/2O3
+SiO.
+Si
+ili
+IV
+Point A
+Point B
+Point C 
+Figure S8 a. Same figure from the main text is copied here for convenience. b. Raman intensity map 
+and the 𝐴1𝑔 peak shift map is given. c. Gate dependent signal from point 4 is subtracted from the 
+gate dependent data from point 2 (red curve) and point 3 (blue curve). Blue curve is multiplied by -1 
+for viewing convenience.  
+ 
+8. Scanning Thermal Microscope Calibration and Seebeck variation estimation 
+The Scanning Thermal Microscope (SThM) measurements were performed on a commercial Bruker 
+Icon instrument with a  VITA-GLA-DM-1 probe. The probe, consisting of the silicon nitride lever with a 
+Pd heater/thermometer has been calibrated on a hot plate to relate the temperature to its electrical 
+resistance. The calibration curves are shown on figure S9.  
+ 
+Figure S9 a. SThM probe calibration of the electrical resistance with the supplied power. b. 
+Temperature as a function of electrical resistance 
+As described elsewhere4,5, the probe is part of a modified Wheatstone bridge which is balanced at low 
+voltage. During the measurements, we applied a combined AC (91 kHz) and DC bias on the bridge 
+which heats the probe and creates a bridge offset that directly measures the probe heater 
+temperature. For most experiments, we applied 1mW on the probe creating a Δ𝑇 of 50 ± 2 K, when 
+the probe was far away from the sample.  
+When the SThM tip is brought into contact with the devices, it locally heats the materials below its 
+apex. While the probe scans the surface, the device open circuit voltage is recorded and amplified via 
+a SR830 voltage preamplifier. This voltage is referred to as the thermovoltage. We excluded any 
+
+E
+Imax
+3(a)
+(b)
+368
+100
+80
+367
+(Ohms)
+60
+366
+40
+365
+R
+ = 363.68 + 12.73 P
+20
+,= -1607.39 + 4.48 R
+applied
+probe
+probe
+364
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+364
+368
+372
+376
+380
+384
+P
+applied (mW)
+R.
+Rprobe (Ohms)shortcut between the probe and the device as no leakage current could be measured between the 
+probe and both contacts. 
+The thermovoltage can be written analytically as6,7,  
+𝑉𝑡ℎ(𝑥) = − ∫ 𝑆(𝑥) 𝜕𝑇
+𝜕𝑥 (𝑥)𝑑𝑥 
+𝐵
+𝐴
+ 
+where 𝑆(𝑥) is the position dependent Seebeck coefficient and 
+𝜕𝑇
+𝜕𝑥 (𝑥) is the position dependent 
+temperature gradient. Both are integrated over the whole device length from A to B.  
+As shown elsewhere6,7, it is possible to deconvolute the Seebeck coefficient from the temperature 
+gradient. This however requires a precise estimation of the temperature gradient and thus the sample 
+temperature rise under the tip, Δ𝑇𝑠𝑎𝑚𝑝𝑙𝑒. 
+As we know the probe temperature far away from the sample (50 ± 2 K) and we monitor its 
+temperature via the Wheatstone bridge, we know that the probe temperature in contact with the 
+sample is 43.8 ± 4 K. The probe cooling occurs because of several heat transfer mechanisms4,5 (solid-
+solid conduction, air conduction, water meniscus, …).  
+For those probes, the Pd heater is however distributed over the whole triangular shaped silicon nitride 
+tip4,5. This implies that the tip temperature and probe temperature are different. We turned to finite 
+element modelling (COMSOL Multiphysics) to estimate the tip temperature over the MoS2 suspended 
+and supported sample. Figure S10 shows the overall simulated probe and sample.  
+We used reported values for the in-plane and out-of-plane MoS2 thermal conductivity as well as for 
+the MoS2-glass interface conductance. Reported values vary greatly in literature8–16. However, to the 
+best of our knowledge, for a thick sample (>10 layers), the values are on the order of 30 Wm-1K-1 for 
+the supported in-plane, 60 Wm-1K-1 for the suspended in-plane and 3 Wm-1K-1 for the cross-plane 
+conductivities. For the substrate interface conductance, we used 1 MWm-2K-1.  
+ 
+ 
+Figure S10 a. Finite element model for the SThM probe on a MoS2 suspended sample. b. Zoomed-in 
+view of the model where the temperature gradient is visible on the sample surface. 
+Using those material parameters, we estimated a ratio between the probe temperature and the tip 
+apex temperature of 4.9. The model also accounts for the tip-sample thermal resistance. This method 
+
+(a)
+(b)and model were experimentally confirmed elsewhere4,5,17. Taking these into consideration, we obtain 
+a sample temperature rise Δ𝑇𝑠𝑎𝑚𝑝𝑙𝑒 of 7.4 ± 0.7 K. This gives a Seebeck variation of 72±10 µVK-1. 
+ 
+References 
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+  
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+arXiv:2301.00744v1  [math.CO]  2 Jan 2023
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A
+GARSIDE MONOID
+THOMAS GOBET AND BAPTISTE ROGNERUD
+Abstract. We study two families of lattices whose number of elements are given by
+the numbers in even (respectively odd) positions in the Fibonacci sequence. The even
+Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially
+ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even
+one. We give a combinatorial proof of the lattice property, relying on a description of
+words for the Garside element in terms of Schröder trees, and on a recursive description
+of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins
+in the lattice. As a byproduct we also obtain that the number of words for the Garside
+element is given by a little Schröder number.
+Contents
+1.
+Introduction
+1
+2.
+Definition and structure of the poset
+3
+2.1.
+Definition of the poset
+3
+2.2.
+Lattice property
+4
+3.
+Schröder trees and words for the Garside element
+7
+3.1.
+labelling of Schröder trees
+7
+3.2.
+Words for the Garside element in terms of Schröder trees
+10
+4.
+Enumerative results
+17
+4.1.
+Number of simple elements
+17
+4.2.
+Number of left-divisors of the lcm of the atoms and odd Fibonacci lattice
+17
+4.3.
+Number of words for the divisors of the Garside element
+18
+References
+20
+1. Introduction
+Several algebraic structures naturally yield examples of lattices: as elementary examples,
+one can cite the lattice of subsets of a given set ordered by inclusion, or the lattice of
+subgroups of a given group.
+One can then study which properties are satisfied by the
+obtained lattices, or conversely, starting from a known lattice, wondering for instance if
+it can be realized in a given algebraic framework, or if a property of the lattice implies
+properties of the attached algebraic structure(s) and vice-versa.
+The aim of this paper is to give a combinatorial description of a finite lattice that
+appeared in the framework of Garside theory. We will not recall results and principles
+of Garside theory as they will not be used in this paper, but the interested reader can
+look at [5, 4] for more on the topic.
+This is a branch of combinatorial group theory
+which aims at establishing properties of families of infinite groups such as the solvability
+of the word problem, the conjugacy problem, the structure of the center, etc. Roughly
+speaking, a Garside group is a group of fraction of a monoid (called a Garside monoid) with
+
+2
+THOMAS GOBET AND BAPTISTE ROGNERUD
+particularly nice divisibility properties, which ensures that the above-mentioned problems
+can be solved. Such a monoid M has no nontrivial invertible element, and comes equipped
+with a distinguished element ∆ (called a Garside element) whose left- and right-divisors are
+finite, coincide, generate the monoid, and form a lattice under left- and right-divisibility.
+The left- or right-divisors of ∆ are called the simples.
+The fundamental example of a Garside group is the n-strand Artin braid group [7].
+It admits several non-equivalent Garside structures (i.e., nonisomorphic Garside monoids
+whose group of fractions are isomorphic to the n-strand braid group), and the lattice of
+simples in the first discovered such Garside structure is isomorphic to the weak Bruhat
+order on the symmetric group. Several widely studied lattices can be realized as lattices
+of simples of a Garside monoid: this includes the lattices of left and right weak Bruhat
+order on any finite Coxeter group [3, 6], the lattice of (generalized) noncrossing partitions
+attached to a finite Coxeter group [1, 2], etc. (see also [12] for many other examples). This
+suggests the following question:
+Question. Which lattices can appear as lattices of simples of Garside monoids ?
+The aim of this paper is to study a family Pn of lattices arising as simples of a family Mn,
+n ≥ 2 of Garside monoids introduced by the first author [8]. For n = 2, the corresponding
+Garside group is isomorphic to the 3-strand braid group B3, while in general it is isomorphic
+to the (n, n + 1)-torus knot group, which for n > 3 is a (strict) extension of the (n + 1)-
+strand braid group Bn+1. The lattice property of Pn follows from the fact proven in op.
+cit. that Mn is a Garside monoid, but it gives very little information about the structure
+and properties of the lattice. For instance, one does not have a formula enumerating the
+number of simples, and only an algorithm to calculate meet and joins in the lattice.
+In Section 2 we give a new proof of the lattice property of Pn (Theorem 2.8) by exhibiting
+the recursive structure of the poset. Every lattice Pn turns out to contain the lattices Pi,
+i < n as sublattices. Note that an ingredient of the proof of Theorem 2.8 is proven later on
+in the paper, as it relies on a combinatorial description for the set of words for the Garside
+element in terms of Schröder trees.
+More precisely, in Section 3 we establish a simple
+bijection between the set of words for ∆n and the set of Schröder trees on n+1 leaves, in such
+a way that applying a defining relation of Mn to a word amounts to applying what we call a
+"local move" on the corresponding Schröder tree (Theorem 3.12 and Corollary 3.13). These
+local moves are given by specific edge contraction and are related to the notion of refinement
+considered in [10]. This allows us to establish in Proposition 3.16 an isomorphism of posets
+between subposets of Pn and Pi, i < n, required in the proof of Theorem 2.8.
+Finally, the obtained recursive description of Pn together with the description of words
+for ∆n in terms of Schröder trees allows us to derive a few enumerative results. This is
+done in Section 4. The first one is that the number of elements of Pn is given by F2n, where
+Fi is the i-th Fibonacci number (Lemma 4.1). We thus call Pn the even Fibonacci lattice.
+The atoms of Mn turn out to have the same left- and right-lcm, which is strictly less than
+∆n. We also show that the sublattice of Pn defined as the order ideal of this lcm has F2n−1
+elements (Lemma 4.3), and thus call it the odd Fibonacci lattice. Other enumerative results
+include the determination of the number of words for the Garside elements (Corollary 3.14),
+and the number of words for the whole set of simples (Theorem 4.7).
+Recall that the Garside monoid Mn under study in this paper has group of fractions
+isomorphic to the (n, n + 1)-torus knot group. This Garside structure was generalized to
+all torus knot groups in [9]. It would be interesting to have a description of the lattices of
+simples of this bigger family of Garside monoids.
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+3
+1
+ρ1
+ρ2
+ρ2ρ1
+ρ3
+ρ1ρ3
+ρ3ρ1
+ρ1ρ3ρ1
+ρ3ρ2
+ρ2ρ1ρ3
+ρ3ρ2ρ1
+ρ2
+3
+ρ3ρ1ρ3
+ρ2
+3ρ1
+(ρ1ρ3)2
+(ρ3ρ1)2
+ρ3
+3
+ρ3ρ2ρ1ρ3
+ρ2
+3ρ1ρ3
+(ρ3ρ1)2ρ3
+ρ4
+3
+Figure 1. The even Fibonacci lattice for n = 3 and (in blue) the odd Fibonacci
+lattice inside it.
+2. Definition and structure of the poset
+2.1. Definition of the poset. The beginning of this section is devoted to explaining how
+the poset under study is defined. We recall the definition of the monoid from which it is
+built, as well as a few properties of this monoid (all of which are proven in [8]).
+Let M be a monoid and a, b ∈ M. We say that a is a left divisor of b (or that b is a
+right multiple of a) if there is c ∈ M such that ac = b. We similarly define right divisors
+and left multiples.
+Let M0 be the trivial monoid and for n ≥ 1, let Mn be the monoid defined by the
+presentation
+(2.1)
+�
+ρ1, ρ2, . . . , ρn
+���� ρ1ρnρi = ρi+1ρn for all 1 ≤ i ≤ n − 1
+�
+.
+We denote by S the set of generators {ρ1, ρ2, . . . , ρn}, and by R the defining relations of
+Mn. This monoid was introduced by the first author in [8, Definition 4.1]. Note that this
+monoid is equipped with a length function λ : Mn −→ Z≥0 given by the multiplicative
+extension of λ(ρi) = i for all i = 1, . . . , n, which is possible since the defining relations
+do not change the length of a word. As a corollary, the only invertible element in Mn is
+
+4
+THOMAS GOBET AND BAPTISTE ROGNERUD
+the identity, and the left- and right-divisibility relations are partial orders on Mn. We will
+write a ≤L b or simply a ≤ b if a left-divides b, and a ≤R b if a right-divides b.
+This monoid was shown to be a so-called Garside monoid (see [8, Theorem 4.18]), with
+corresponding Garside group (which has the same presentation as Mn) isomorphic to the
+(n, n + 1)-torus knot group, that is, the fundamental group of the complement of the
+torus knot Tn,n+1 in S3.
+Garside monoids have several important properties.
+Among
+them, the left- and right-divisibility relations equip Mn with two lattice structures, and
+Mn comes equipped with a distinguished element ∆n, called a Garside element, which has
+the following two properties
+(1) The set of left divisors of ∆n coincides with its set of right divisors, and forms a
+finite set.
+(2) The set of left (or right) divisors of ∆n generates Mn.
+This Garside element is given by ∆n = ρn+1
+n
+. In particular, as any Garside monoid is a
+lattice for both left- and right-divisibility, the set Div(∆n) of left (or right) divisors of ∆n
+is a finite lattice if equipped by the order relation given by the restriction of left- (or right-)
+divisibility on Mn. The set Div(∆n) is the set of simple elements or simples of Mn. In
+general (Div(∆n), ≤L) and (Div(∆n), ≤R) will not be isomorphic as posets. But we always
+have
+(Div(∆n), ≤L) ∼= (Div(∆n), ≤R)op
+(see for instance [8, Lemma 2.19]; such a property holds in any Garside monoid).
+We will give a new proof that (Div(∆n), ≤) (and hence (Div(∆n), ≤R) is a lattice, in
+a way which will exhibit a recursive structure of the poset. To this end, we will require
+(sometimes without mentioning it) a few basic results on the monoid Mn which are either
+explained above or proven in [8]:
+(1) The left- and right-divisibility relations on Mn are partial orders.
+(2) The monoid Mn is both left- and right-cancellative, i.e., for a, b, c ∈ Mn, we have
+that ab = ac ⇒ b = c, and ba = ca ⇒ b = c (see [8, Propositions 4.9 and 4.12]),
+(3) The set of left- and right-divisors of ∆n coincide. In fact, the element ∆n is central
+in Mn, hence as Mn is cancellative, for a, b ∈ Mn such that ab = ∆n, we have
+ab = ba (see [8, Proposition 4.15])
+2.2. Lattice property. The aim of this subsection is to prove a few properties of simple
+elements of Mn, and to derive a new algebraic proof that Div(∆n) is a lattice.
+Proposition 2.1. Let x1x2 · · · xk be a word for ∆n, with xi ∈ S for all i = 1, . . . , k. There
+are i1 = 1 < i2 < · · · < iℓ ≤ k such that
+• For all j = 1, . . . , ℓ, the word yj := xijxij+1 · · · xij+1−1 (with the convention that
+iℓ+1 = k + 1) is a word for a power of ρn,
+• The decomposition y1|y2| · · · |yℓ of the word x1x2 · · · xk is maximal in the sense that
+no word among the yj can be decomposed as a product of two nonempty words which
+are words for powers of ρn.
+Morever, a decomposition with the above properties is unique.
+Proof. The existence of the decomposition is clear using the fact that Mn is cancellative:
+given the word x1x2 · · · xk, consider the smallest i ∈ {1, 2, . . . , k} such that x1x2 · · · xk is
+a word for a power of ρn. Such an i has to exist, as x1x2 · · · xk is a word for a power of
+ρn. Then set i2 := i + 1. By cancellativity in Mn, since x1 · · · xi and x1 · · · xk are both
+words for a power of ρn, the word xi+1 · · · xk must also be a word for a power of ρn. Hence
+one can go on, arguing the same with the word xi+1 · · · xk. Again by cancellativity, this
+decomposition must be maximal.
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+5
+Now assume that the decomposition is not unique, that is, assume that y1|y2| · · · |yℓ
+and z1|z2| · · · |zℓ′ are two decompositions of the word x1x2 · · · xk satisfying the properties
+of the statement. As both y1 and z1 are words for a power of ρn, if y1 ̸= z1, then one
+word must be strict prefix of the other, say z1 is a strict prefix of y1. But this contradicts
+the maximality of the decomposition y1|y2| · · · |yℓ: indeed if y1 = x1x2 · · · xi2−1 and z1 =
+x1x2 · · · xp with p < i2 − 1, we can decompose y1 nontrivially as x1x2 · · · xp|xp+1 · · · xi2−1,
+and by cancellativity both x1 · · · xp and xp+1 · · · xi2−1 are words for powers of ρn.
+□
+Example 2.2. Consider the word ρ3ρ1ρ7ρ1ρ7ρ5ρ4ρ7ρ7ρ1ρ7ρ6 in M7. We claim that this is a
+word for the Garside element ρ8
+7 of M7. Indeed, using the defining relation ρ1ρ7ρi = ρi+1ρ7
+with i = 5 and 6, we get that
+ρ3(ρ1ρ7ρ1ρ7ρ5)ρ4ρ7ρ7(ρ1ρ7ρ6) = ρ3ρ3
+7ρ4ρ4
+7,
+and we observe also applying defining relations that
+ρ3ρ3
+7ρ4ρ4
+7 = ρ1ρ7ρ2ρ2
+7ρ4 = ρ1ρ7ρ1ρ7ρ1ρ7ρ4 = ρ1ρ7ρ1ρ7ρ5ρ7 = ρ1ρ7ρ6ρ2
+7 = ρ4
+7.
+The decomposition according to Proposition 2.1 is given by
+ρ3ρ1ρ7ρ1ρ7ρ5ρ4
+�
+��
+�
+:=y1
+| ρ7
+����
+:=y2
+| ρ7
+����
+:=y3
+| ρ1ρ7ρ6
+� �� �
+:=y4
+.
+It is indeed clear by considering λ(u) for u prefixes of y1 or y4 that whenever u is a proper
+prefix, we do not have λ(u) equal to a multiple of 7, which is a necessary condition for a
+word to represent a power of ρ7.
+Lemma 2.3. Let 1 ≤ k ≤ n. Then
+S ∩ {x ∈ Div(∆n) | x ≤ ρkρk
+n} = {ρ1, ρ2, . . . , ρk}.
+Proof. We argue by induction on k. The result is clear for k = 1, as no defining relation of
+Mn can be applied to the word ρ1ρn. Now let k > 1. Observe that
+ρkρk
+n = (ρ1ρn)k = (ρ1ρn)(ρ1ρn)k−1.
+In particular we have ρ1 ≤ ρkρk
+n and by induction, we get ρ1ρnρi ≤ ρkρk
+n for all 1 ≤ i ≤ k−1.
+As ρ1ρnρi = ρi+1ρn we get that {ρ1, ρ2, . . . , ρk} ⊆ S ∩ {x ∈ Div(∆n) | x ≤ ρkρk
+n}.
+It remains to show that no other ρi can be a left-divisor of ρkρk
+n. Hence assume that
+i > k and ρi ≤ ρkρk
+n. Hence there is a word x1x2 · · · xp for ρkρk
+n, where xi ∈ S for all i,
+such that x1 = ρi. As the words x1x2 · · · xp and ρkρk
+n represent the same element, they
+can be related by a finite sequence of words w0 = x1x2 · · · xp, w1, . . . , wq = ρkρk
+n, where
+each wi is a word with letters in S and wi+1 is obtained from wi by applying a single
+relation somewhere in the word. As the first letter of w0 differs from the first letter of
+wq, there must exist some 0 ≤ ℓ < q such that wℓ begins by ρi but wℓ+1 does not. It
+follows that the relation allowing one to pass from wℓ to wℓ+1 has to be applied at the
+beginning of the word wℓ. But the only possible relation with one side beginning by ρi
+is ρiρn = ρ1ρnρi−1. It follows that ρ1ρnρi−1 ≤ ρkρk
+n = (ρ1ρn)k. By cancellativity, we get
+that ρi−1 ≤ (ρ1ρn)k−1 = ρk−1ρk−1
+n
+. By induction this forces one to have i − 1 ≤ k − 1,
+contradicting our assumption that i > k.
+□
+Similarly, we have
+Lemma 2.4. Let 1 ≤ k ≤ n. Then
+S ∩ {x ∈ Div(∆n) | x ≤R ρk
+n} = {ρn, ρn−1, . . . , ρn−k+1}.
+
+6
+THOMAS GOBET AND BAPTISTE ROGNERUD
+Proof. As for Lemma 2.3, we argue by induction on k. The result is clear for k = 1. Hence
+assume that k > 1. As (ρ1ρn)n−jρj = ρn−j+1
+n
+, we get that ρj ≤R ρk
+n for all j such that
+n − j + 1 ≤ k, that is, for all j ≥ n − k + 1. It remains to show that no other ρj can
+right-divide ρk
+n. Hence assume that ρj ≤R ρk
+n, where j < n − k + 1. Arguing as in the
+proof of Lemma 2.3, we see that ρ1ρnρj = ρj+1ρn must be a right-divisor of ρk
+n, hence by
+cancellativity that ρj+1 ≤R ρk−1
+n
+. By induction this forces j + 1 ≥ n − k + 2, contradicting
+our assumtion that j < n − k + 1.
+□
+For x ∈ Div(∆n), let d(x) := max{k ≥ 0 | ρk
+n ≤ x}. Let 0 ≤ i ≤ n + 1 and let
+Di
+n := {x ∈ Div(∆n) | d(x) = i}.
+Note that
+Div(∆n) =
+�
+0≤i≤n+1
+Di
+n.
+We have Dn
+n = {ρn
+n}, Dn+1
+n
+= {∆n}.
+Lemma 2.5. Let x ∈ Div(∆n) and i = d(x). Let x′ ∈ Mn such that x = ρi
+nx′. Note that
+x′ ∈ D0
+n. Let x1x2 · · · xk be a word for x, where xi ∈ S for all i = 1, . . . , k. Then there is
+1 ≤ ℓ ≤ k such that x1x2 · · · xℓ is a word for ρi
+n (and hence xℓ+1 · · · xk is a word for x′ by
+cancellativity). In other words, any word for x has a prefix which is a word for ρi
+n.
+Proof. It suffices to show that if z1z2 · · · zp is an expression for x such that z1z2 · · · zq is
+an expression for ρi
+n (q ≤ p, then one cannot apply a defining relation of Mn on the word
+z1z2 · · · zp simultaneously involving letters of the word z1z2 · · · zq and letters of the word
+zq+1 · · · zp. Let us consider the three possible cases where this could occur: one could have
+ρ1ρn|ρj, ρ1|ρnρj, or ρj+1|ρn (1 ≤ j < n), where the | separates the letters zq and zq+1.
+The last two cases cannot happen, since one would have zq+1 = ρn, hence zq+1 · · · zp would
+be a word for x′ beginning by ρn, contradicting the fact that x′ ∈ D0
+n. It remains to show
+that the case ρ1ρn|ρj cannot happen. Hence assume that zq−1 = ρ1, zq = ρn, zq+1 = ρj.
+By cancellativity, as z1z2 · · · zq is a word for ρi
+n, it implies that ρ1 ≤R ρi−1
+n
+. By lemma 2.4,
+this implies that n − (i − 1) + 1 = 1, hence that i = n + 1.
+Since x ∈ Div(∆n) and
+x = ρn+1
+n
+x′ = ∆nx′, we get x′ = 1, contradicting the fact that zq+1 = ρj.
+□
+Lemma 2.6. Let i, j ∈ {0, 1, . . . , n + 1}, with i ̸= j. Let x ∈ Di
+n, y ∈ Dj
+n. Assume that
+x ≤ y. Then i < j and x < ρj
+n ≤ y.
+Proof. It is clear that i < j, since ρi
+n ≤ y as ρi
+n ≤ x, hence j < i would contradict y ∈ Dj
+n.
+In particular x < y. Let x′, y′ such that x = ρi
+nx′ and y = ρj
+ny′. Note that x′, y′ both lie
+in D0
+n. Since x ≤ y and Mn is cancellative, we get that x′ < ρj−i
+n y′. It implies that there
+exists a word x1x2 · · · xk for ρj−i
+n
+y′ (xi ∈ S) and 1 ≤ ℓ < k such that x1x2 · · · xℓ is a word
+for x′. Now by lemma 2.5, there is 0 ≤ ℓ′ ≤ k such that x1x2 · · · xℓ′ is a word for ρj−i
+n . If
+ℓ′ ≤ ℓ, then ρj−i
+n
+≤ x′, contradicting the fact that x′ ∈ D0
+n. Hence ℓ′ > ℓ, and x′ < ρj−i
+n .
+Multiplying by ρi
+n on the left we get x < ρj
+n.
+□
+Lemma 2.7. Let z1, z2 ∈ Di
+n. Let 1 ≤ k1 < k2 ≤ n and assume that there are two cover
+relations z1 ≤· ρk1
+n , z2 ≤· ρk2
+n in (Div(∆n), ≤). Then z1 < z2.
+Proof. As z1 ≤ ρk1
+n , z2 ≤ ρk2
+n
+are cover relations, there are 1 ≤ j1, j2 ≤ n such that
+z1ρj1 = ρk1
+n , z2ρj2 = ρk2
+n . By lemma 2.4, for ℓ ∈ {1, 2} we have jℓ ∈ {n − kℓ + 1, . . . , n} and
+ρkℓ
+n = ρjℓ+kℓ−1−n
+n
+(ρ1ρn)n−jℓρjℓ.
+In particular, we have
+zℓ = ρjℓ+kℓ−1−n
+n
+(ρ1ρn)n−jℓ
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+7
+and as (ρ1ρn)n−jℓ = ρn−jℓρn−jℓ
+n
+, by lemma 2.3 we see that ρn cannot be a left divisor of
+(ρ1ρn)n−jℓ, and hence that d(zℓ) = jℓ + kℓ − 1 − n. But d(zℓ) = i for ℓ ∈ {1, 2}, and since
+k1 < k2 we deduce that j1 > j2. Since zℓ = ρi
+n(ρ1ρn)n−jℓ, we get that z1 < z2, which
+concludes the proof.
+□
+Theorem 2.8. The poset (Div(∆n), ≤) is a lattice. Given i, j ∈ {0, 1, . . . , n+1} with i ≤ j
+and x ∈ Di
+n, y ∈ Dj
+n, we have
+x ∧ y = x ∧i
+��
+i
+{z ∈ Di
+n | z ≤ y}
+�
+,
+where ∨i and ∧i denote the meet and join on the restriction of the left-divisibility order on
+Di
+n, which itself forms a lattice. Note that if i = j we simply get x ∧ y = x ∧i y.
+Proof. The proof is by induction on n. We have Div(∆0) = {•}, and Div(∆1) = {1, ρ1, ρ2
+1},
+which is a lattice. Hence assume that n ≥ 2. By Proposition 3.16 below, the restriction of
+the left-divisibility to Di
+n yields an isomorphism of poset with Div(∆n−i) if i ̸= 0, n+1, while
+the restriction to D0
+n yields an isomorphism of poset with Div(∆n−1), and the restriction to
+Dn+1
+n
+an isomorphism of posets with Div(∆0) = {•}. In particular, by induction, all these
+posets are lattices. As the poset (Div(∆n), ≤) is finite and admits a maximal element, it
+suffices to show that x ∧ y as defined by the formula above is indeed the join of x and y.
+It is clear that x∧y ≤ x. Let us show that x∧y ≤ y. If i = j this is clear, hence assume
+that i < j. By lemma 2.6 we see that
+�
+i
+{z ∈ Di
+n | z ≤ y} =
+�
+i
+{z ∈ Di
+n | z ≤ ρj
+n}.
+It suffices to check that �
+i{z ∈ Di
+n | z ≤ ρj
+n} ≤ ρj
+n. Note that
+�
+i
+{z ∈ Di
+n | z ≤ ρj
+n} =
+�
+i
+{z ∈ Di
+n | z ≤ ρj
+n and (z ≤· x ≤ ρj
+n ⇒ x /∈ Di
+n)}.
+Now by lemma 2.6, if z ∈ Di
+n and x is any element such that z ≤· x ≤ ρj
+n and x /∈ Di
+n, then
+x = ρk
+n for some k (necessarily smaller than or equal to j). It implies that
+�
+i
+{z ∈ Di
+n | z ≤ ρj
+n} =
+�
+i
+{z ∈ Di
+n | z ≤ ρj
+n and z ≤· ρk
+n for some k ≤ j}.
+By lemma 2.7, we have that
+�
+i
+{z ∈ Di
+n | z ≤ ρj
+n and z ≤· ρk
+n for some k ≤ j}
+has to be an element of the set {z ∈ Di
+n | z ≤ ρj
+n and z ≤· ρk
+n for some k ≤ j}, hence that
+it is in particular a left-divisor of ρj
+n (and hence of y).
+Now assume that u ≤ x, y. We can assume that u ∈ Di
+n, otherwise by lemma 2.6 we
+have u < ρi
+n ≤ x ∧ y. As u ≤ y, we have that u ≤ �
+i{z ∈ Di
+n | z ≤ y}. And hence, that
+u ≤ x ∧i
+��
+i{z ∈ Di
+n | z ≤ y}
+�
+= x ∧ y.
+□
+3. Schröder trees and words for the Garside element
+3.1. labelling of Schröder trees. A rooted plane tree is a tree embedded in the plane
+with one distinguished vertex called the root. The vertices of degree 1 are called the leaves
+of the tree and the other vertices are called inner vertices. One can consider rooted trees
+as directed graphs by orienting the edges from the root toward the leaves. If there is an
+oriented edge from a vertex v to a vertex w, we say that v is the parent of w and w is a
+child of v. As can be seen in Figure 2, we draw the trees with their root on the top and the
+
+8
+THOMAS GOBET AND BAPTISTE ROGNERUD
+leaves on the bottom. The planar embedding induces a total ordering (from left to right)
+on the children of each vertex, hence we can speak about the leftmost child of a vertex.
+Alternatively one has a useful recursive definition of a rooted plane tree: it is either the
+empty tree with no inner vertex and a single leaf or a tuple T = (r, Tr) where r is the root
+vertex and Tr is an ordered list of rooted plane trees. If T is a tree with the first definition,
+the vertex r is its root and the list Tr is the list of subtrees, ordered from left to right,
+obtained by removing the root r and all the edges adjacent to r in T.
+Definition 3.1.
+(1) A Schröder tree is a rooted plane tree in which each inner vertex has at least two
+children.
+(2) A binary tree is a rooted plane tree in which each inner vertex has exactly two
+children.
+(3) The size of a tree is its number of leaves.
+(4) The height of a tree is the number of vertices in a maximal chain of descendants.
+(5) The Schröder tree on n leaves in which every child of the root is a leaf is called
+the Schröder bush. We denote it by δn.
+(6) The Schröder tree given by the binary tree in which every right child (resp. every
+left child) is a leaf is called a left comb (resp. a right comb).
+· · ·
+Figure 2. From left to right: the unique Schröder tree with 1 leaf, the unique
+Schröder tree with two leaves, the three Schröder trees with 3 leaves.
+Then the
+Schröder bush and on its right a left comb.
+The Schröder trees are counted by the so-called little Schröder numbers. The sequence
+starts with 1, 1, 3, 11, 45, 197, 903, 4279, 20793, ... and is referred as A001003 in [11].
+We will label (and read the labels of) the vertices and the leaves of our trees using the
+so-called post-order traversal. This is a recursive algorithm that visits each vertex and leaf
+of the tree exactly once. Concretely, if T =
+�
+r, (T1, . . . , Tk)
+�
+is a rooted planar tree, then
+we recursively apply the algorithm to T1, T2 until Tk and finally we visit the root r. When
+the algorithm meets an empty tree it visits its leaf and then, the recursion stops and it
+goes up one level in the recursive process. The first vertex visited by the algorithm is the
+leftmost leaf of T, then the algorithm moves to its parent v (but does not visit v) and visits
+the second subtree of v starting with the leftmost leaf and so on. We refer to Figure 3 for
+an illustration where the first vertex visited by the algorithm is labeled by 1, the second
+by 2 and so on. The last vertex visited by the algorithm is always the root of T. Let m, n
+be two integers such that m ≥ n − 1. We then label a Schröder tree T with n ≥ 2 leaves
+by labelling its vertices one after the other with respect to the total order defined by the
+post-order traversal, using the following rules:
+(1) Let v be the leftmost child of a vertex w. Then w is the root of a Schröder tree
+�
+w, (T1, · · · , Tk)
+�
+and v is the root of T1. The label λ(v) of v is equal to the number
+of leaves of the forest consisting of all the trees T2, · · · , Tk.
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+9
+20
+11
+3
+1
+2
+6
+4
+5
+10
+7
+8
+9
+12
+19
+15
+13 14
+18
+16 17
+Figure 3. Post-order traversal of a Schröder tree of size 12.
+(2) If v is not the leftmost child of a vertex of T, we consider LD(v) the set of its
+leftmost descendants consisting of the leftmost child of v and its leftmost child and
+so one. Then the label of v is m − �
+w∈LD(v) λ(w). Note that using the post-order
+traversal, the label of the leftmost descendants of a vertex v are already determined
+when we visit v.
+The result is a labelled Schröder tree that we denote by Lm(T).
+This procedure is
+illustrated in Figure 4.
+Definition 3.2. Let Lm(T) be a labelled Schröder tree with n leaves labelled by m ≥ n−1.
+The sum of the labels of the vertices of T is called its weight (with respect to m).
+Lemma 3.3. Let T be a Schröder tree with n leaves and m ≥ n − 1. Then the integers
+labelling Lm(T) are strictly nonnegative with the exception of the root which may be labelled
+by 0.
+Proof. If a vertex is a leftmost child, then its label is a number of leaves, hence it is positive.
+If v is not a leftmost child, then it is labelled by m − �
+w∈LD(v) λ(w). Each λ(w) is equal
+to a certain number of leaves of T and the set of leaves associated to distinct vertices of
+LD(v) do not intersect. Moreover, exactly one element of LD(v) is a leaf and this leaf is
+not counted in �
+w∈LD(v) λ(w). We therefore have
+(3.1)
+
+
+�
+w∈LD(v)
+λ(w)
+
+ + 1 ≤ n,
+hence m − �
+w∈LD(v) λ(w) ≥ 0. Moreover if �
+w∈LD(v) λ(w) = m, then by (3.1) we must
+have m = n − 1. It follows that v has n descendants since the leftmost leaf which is a
+descendant of v is not counted, hence v is the root of T.
+□
+This labelling is almost determined by the recursive structure of the tree, as shown by
+the following result.
+Lemma 3.4. Let T =
+�
+r, (T1, . . . , Tk)
+�
+be a Schröder tree and v be a vertex of Ti for
+i ∈ {1, . . . , k}. Then,
+(1) If v is not the root of T1, then its label in Lm(T) is equal to its label in Lm(Ti).
+(2) If v is the root of T1, then its label in Lm(T1) is equal to the sum of the labels of v
+and of the root of T in Lm(T).
+Proof. Let v be a vertex of Ti. If v is a leftmost child in T which is not the root of T1,
+then its label is a number of leaves of a certain forest which is contained in Ti. Hence this
+number is the same in the big tree T or in the extracted tree Ti. If v is not a leftmost
+child, then its label is determined by the labels of its leftmost descendants, hence it is the
+same in the tree T as in the extracted tree Ti since we have just shown that the labels of
+leftmost descendants which are not the root of T1 agree. The root of T1 has a different
+behaviour since in T it is a leftmost child and this is not the case in T1. Hence if v is the
+root of T1, denoting by λ1 the label of v in T1, we have λ1(v) = m − �
+w∈LD(v) λ1(w). The
+
+10
+THOMAS GOBET AND BAPTISTE ROGNERUD
+labels of the descendants of v are the same in T and in T1, that is, we have λ1(w) = λ(w)
+for all w ∈ LD(v). In T, the label of the root r is given by
+λ(r) = m − λ(v) −
+�
+w∈LD(v)
+λ(w) = m − λ(v) −
+�
+w∈LD(v)
+λ1(w).
+Hence we have λ(r) + λ(v) = λ1(v).
+□
+3.2. Words for the Garside element in terms of Schröder trees. Reading the la-
+belled tree Lm(T) using the post-order traversal and associating the generator ρi to the
+letter i with the convention that ρ0 = e, gives a map Φm from the set of Schröder trees
+labelled by m to the set S⋆ of words for the elements of the monoid Mm. We refer to
+Figure 4 for an illustration.
+0
+5
+5
+1
+11
+10
+1
+11
+9
+2
+11 11
+11
+8
+2
+1
+11
+10
+1
+11
+Figure
+4. Example
+of
+the
+labelling
+of
+a
+Schröder
+tree
+of
+size
+12
+with
+m
+=
+11.
+The
+corresponding
+element
+in
+the
+monoid
+M11
+is
+ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5ρ11ρ1ρ11ρ2ρ1ρ11ρ10ρ8.
+Definition 3.5. Let T be a non-empty Schröder tree. If T has a subtree T1 satisfying the
+three following properties:
+(1) The root r1 of T1 is not the root of T, hence it has a parent r0 which has at least
+two children,
+(2) The root r1 has exactly two children,
+(3) The right subtree of T1 is the empty tree with only one leave.
+Then, we can construct another tree �T by contracting the edge r0 − r1, in other words by
+removing the root r1 of T1 and attaching the two subtrees of T1 to r0. See Figure 5 for an
+illustration. We call such a transformation, or the inverse transformation, a local move.
+Note that, since r0 has at least two children in the configuration described above (see also
+the left picture in Figure 5), we get that r0 has at least three children in the configuration
+obtained after applying the local move. In particular, to apply a local move in the other
+direction, we need to have a Schröder tree �T with a subtree T1 satisfying :
+(1) The parent r0 of T1 (which is allowed to be the root of T) has at least three children,
+(2) The tree T1 is not the last child of r0, and is directly followed by an empty tree
+with only one leaf.
+r0
+Sk
+r1
+r2
+A1
+Sk+2
+←→
+r0
+Sk
+r2
+A1
+Sk+2
+Figure 5. Local move.
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+11
+Formally, if the subtree of T with root r0 is S =
+�
+r0, (S1, · · · , Sk, T1, Sk+2, · · · , Sr)
+�
+and
+the subtree T1 is
+�
+r1, (A1, A2)
+�
+, then we obtain the tree �T by replacing S by
+�
+r0, (S1, · · · , Sk, A1, A2, Sk+2, · · · , Sr)
+�
+.
+Lemma 3.6. Let T and S be two Schröder trees with n leaves. Then one can pass from
+the tree T to the tree S by applying a sequence of local moves.
+Proof. It is enough to show that T can be transformed into the Schröder bush δn–recall
+that this is the Schröder tree in which every child of the root is a leaf–by a sequence of
+local moves. The Schröder tree S can then be transformed as well into δn, and hence T
+can be transformed into S. We argue by induction on the number of leaves. For n = 1 and
+n = 2 there is nothing to prove. If T has a subtree S which is not of the form δk, then we
+can transform S into δk for some k by applying the induction hypothesis to S. Hence we
+can assume that T =
+�
+r, (δn1, · · · , δnk)
+�
+with � ni = n. If T is not equal to δn, then it has
+at least a non-empty subtree S. If S has only two leaves, then we can apply a local move to
+remove its root and to attach the two leaves to the root of T. If it has more than 3 leaves,
+by induction there is a sequence of local moves from S to a left comb. Then, by repeatedly
+applying a local move at the root of the left comb, we remove all the inner vertices of the
+left comb and attach all its leaves to the root of T. Applying this to all subtrees S of T
+wich are not empty, we end up getting δn.
+□
+Lemma 3.7. Let T be a Schröder tree with n leaves and m ≥ n − 1. Then Φm(T) is a
+word for ρn−1(ρm)n−1ρm−n+1 in Mm. In particular if m = n − 1, then it is a word for the
+Garside element of Mn−1.
+Proof. If T = δn is the Schröder tree with only one root and n leaves, then Φm(T) =
+ρn−1(ρm)n−1ρm−n+1. If T is another Schröder tree, then by Lemma 3.6 there is a sequence
+of local moves from T to δn. To finish the proof it is enough to show that applying a local
+move to a Schröder tree T amounts to applying a relation of the monoid Mm to Φm(T).
+This is easily obtained by staring at Figure 5.
+Indeed, if T is the tree at the left of Figure 5, then the label of r2 is 1, the label of the
+leaf on its right is m and the label of r1 is a certain integer ℓ. Since r1 is not the root of
+T, we have 1 ≤ ℓ. Moreover, since r1 is not a leaf of T, we have ℓ < m. Hence in Φm(T)
+we have the factor ρ1ρmρℓ with 1 ≤ ℓ ≤ m − 1.
+If �T denotes the right tree of Figure 5, then the label of r2 is ℓ + 1. Indeed r2 is a
+leftmost child in �T if and only if r1 is a leftmost child in T. In this case its label is the
+number of leaves of the forest in its right and in �T there is precisely one more leaf in this
+forest than in T. In the other case, the label of r2 in �T is m − �
+w∈LD(r2) λ(w). The label
+of r1 is ℓ = m − 1 − �
+w∈LD(r2) λ(w). So the label of r2 is ℓ + 1. The leaf on the right of r2
+in �T is labelled by m, hence Φm( �T) is obtained by replacing ρ1ρmρℓ in Φm(T) by ρℓ+1ρm,
+and vice-versa.
+□
+Proposition 3.8. For m = n−1, the map Φm from the set of Schröder trees with n leaves
+to the set of words for ρn
+n−1 in Mn−1 is surjective.
+Proof. We have to show that to each word y for ρn
+n−1 ∈ Mn−1, we can attach a Schröder
+tree T with n leaves, in such a way that Φm(T) = y. The word y and the word ρn
+n−1
+can be transformed into each other by applying a sequence of defining relations of Mm.
+We already know that the word ρn
+n−1 is in the image of Φm since it is the image of the
+Schröder bush. To conclude the proof, we therefore need to show the following claim: given
+a Schröder tree S, if the corresponding labelling has a substring of the form 1mℓ (resp.
+(ℓ + 1)m) with 1 ≤ ℓ ≤ m − 1, then we are necessarily in the configuration of the left
+
+12
+THOMAS GOBET AND BAPTISTE ROGNERUD
+picture in Figure 5 (resp. the right picture), and hence we can apply a local move. Indeed,
+as one can pass from the word ρn
+n−1 to the word y by a sequence of defining relations
+let y0 = ρn
+n−1, y1, . . . , yk = y be expressions of ρn
+n−1 such that yi is obtained from yi−1
+by applying a single relation in Mm.
+Applying the relation on y0 = Φm(T) to get y1
+corresponds to applying a local move on T to get a Schröder tree T1 and as seen in the
+proof of lemma 3.7, we get Φm(T1) = y1.
+To show the claim, assume that S is a Schröder tree with labelling having a substring
+of the form 1mℓ with 1 ≤ ℓ ≤ m − 1. Note that m can only be the label of a leaf. Let v be
+the parent of that leaf. It is a root of a family of trees, say (v, T1, . . . , Tk) and our leaf with
+label m corresponds to one of the trees Ti (which has to be empty). It is clear that such a
+tree cannot be T1: indeed, as T1 is the leftmost child of v, in that case m = n − 1 would
+be the number of leafs in the forest T2, . . . , Tk, which is at most n − 1. As m = n − 1,
+the only possibility would be that v is the root of S, hence m would be the first label and
+therefore could not be preceded by a label 1. Hence m labels one of the trees T2, . . . , Tk,
+say Ti. It follows that the label 1 preceding m is the label of the root of Ti−1. If i = 2
+then k = 2 as the label 1 is then the label of the leftmost child of v, meaning that there is
+only one leaf in the forest T2, . . . , Tk. In that case, it only remains to show that v cannot
+be the root of S to match the configuration in the left picture of Figure 5. But this is
+clear for if v was the root of S, the last label would be m corresponding to T2, hence
+no ℓ could appear. Hence v is not the root of S, and its label is ℓ. Now if i ̸= 2, then
+i − 1 ̸= 1. The root v′ of Ti−1 is labelled by 1 and as v′ is not the leftmost child of v, we
+have 1 = λ(v′) = m−�
+w∈LD(v′) λ(w), yielding �
+w∈LD(v′) λ(w) = m−1. This means that
+there are m leaves in Ti−1, and as there is one leaf in Ti and m = n−1, the only possibility
+is that i − 1 = 1 and k = 2, contradicting i ̸= 2.
+Now, assume that S is a Schröder tree with labelling having a substring of the form
+(ℓ + 1)m with 1 ≤ ℓ ≤ m − 1. Again, m can only label a leaf. Let v be the parent of that
+leaf as above, which is a root of a family T1, . . . , Tk of trees with Ti corresponding to our
+leaf for some i. We need to show that i ̸= 1 and k ≥ 3. In the previous case we have seen
+that if i = 1, then m = n − 1 is the number of leaves in T2, . . . , Tk, forcing v to be the root
+of S and m to be the first label in S. Hence i ≥ 2. If k = 2 (hence i = 2), then the root of
+T1 is labelled by 1 = ℓ + 1, contradicting 1 ≤ ℓ. Hence k ≥ 2.
+□
+Lemma 3.9.
+(1) Let T be a Schröder tree with n leaves labelled by m ≥ n − 1. Then,
+the weight of T is nm.
+(2) Let w be a vertex of T which is not a leaf and v its leftmost child, that is w is the
+root of a Schröder tree
+�
+w, (T1, · · · , Tk)
+�
+and v is the root of T1. Then the weight of
+the forest F = (T2, · · · , Tk) attached to w is λ(v)m, and the labelling of of a vertex
+in a tree Ti for i ≥ 2 is the same as its labelling inside T.
+Proof. The first result is proved by induction on the number of leaves. If the tree has one
+leaf the result holds by definition of our labelling. Let T = (r, T1, · · · , Tk) be a Schröder
+tree, where Ti has ni leaves. By induction, the tree Ti has weight mni for i ≥ 1. Us-
+ing Lemma 3.4, the sum of the labels of the vertices of the tree Ti (in T) is equal to mni
+for i ≥ 2 and the sum of the labels of the vertices of T1 and of the root of T is equal to
+mn1. Hence, the tree T has weight �k
+i=1 mni = mn. For the second point, the number of
+leaves of the forest F is equal to λ(v). Hence by the first point, the forest F has weight
+λ(v)m.
+□
+Proposition 3.10. Let m ≥ n − 1. Then the map Φm from the set of Schröder trees with
+n leaves to the set of words for the element ρn−1(ρm)n−1ρm−n+1 ∈ Mm is injective.
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+13
+Proof. Let T = (r, T1, · · · , Tk) be a Schröder tree with n leaves labelled by m. This proof is
+purely combinatorial and it only involves the word W in N obtained by reading the labels
+of the tree in post-order. The first step of the proof is to remark that one can recover the
+decomposition ‘root and list of subtrees’ of a Schröder tree just by looking at W. We will
+illustrate the algorithm in Example 3.11 below. Precisely we want to split the word W
+into a certain number of factors W = W1 · · · Wk such that each subword Wi is equal to the
+word obtained by reading the labels of Lm(Ti) in post-order.
+The first letter w1 of W is the label of the leftmost leaf r1 of T and by induction we will
+find the letters w2, w3, · · · , wi corresponding to the ancestors r2, r3, · · · , ri of r1. Since the
+labels of these vertices count a number of leaves of T, when �i
+j=1 wj = n − 1, then all the
+leaves of T have been counted so ri is the root of T1 and we stop the induction.
+If we have found the letter wk corresponding to rk ̸= r, then wk is the number of leaves
+of the right forest attached to the parent rk+1 of rk. By Lemma 3.9, the weight of F is
+m · wk, hence the word obtained by reading the vertices of F is wk+1 · · · wi where i is the
+smallest integer such that �i
+j=k+1 wj = mwk. All these letters correspond to the vertices
+of F, hence the next letter is the label of the vertex read after F in the post-order traversal,
+which is the vertex rk+1.
+Since the word W only contains strictly non-negative integers (except possibly the label
+of the root of T), at each step of the induction the value w1 +· · ·+wi strictly increases and
+the induction stops. If wn1 is the letter corresponding to the leftmost child of the root of T,
+then the word w1 · · · wn1 is the word obtained by reading all the vertices of the subtree T1.
+By Lemma 3.4, this is almost the word obtained by reading Lm(T1) we just need to ‘correct’
+the label of the root of T1 by adding the label of the root of T which is the last letter wl of
+W. To conclude the word consisting of the labels of T1 is WT1 = w1 · · · wn1−1(wn1 + wl).
+Let �
+W be the word obtained by removing the letters w1, · · · , wn1 and wl. We use the
+same procedure to extract the subwords corresponding to the other subtrees of T. Due
+to the asymmetry of Lemma 3.4, there is a slight difference. We have found all the labels
+w1, w2, · · · , wt of the vertices r1, r2, · · · , rt of the left branch of Ti when �t
+j=1 wj = m and
+there is no need to ‘correct’ the word as above.
+We are now ready to prove that Φm is injective.
+If the words of two trees T =
+(r, (T1, · · · , Tk)) and S = (s, (S1, · · · , Sl)) obtained by reading the labels of their vertices
+in post-order are equal, then by the discussion above we have k = l and for i ∈ {1, · · · , k},
+the words obtained by reading the vertices of the subtrees Lm(Ti) and Lm(Si) are equal.
+By induction on the number of leaves, we have Si = Ti for i = 1, · · · , k and we get that
+T = S.
+□
+Example 3.11. We illustrate the decomposition involved in the proof of Proposition 3.10
+with the example of Figure 4. We consider the leftmost subtree T1 of T with n = 7 leaves
+and which is labelled by m = 11. We have Φ11(T1) = ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5. The
+first letter 1 tels us that the forest on the right of the leftmost leaf r1 has 1 vertex. Its
+weight is m = 11. Hence ρ11 labels the only vertex of the forest and the next letter 5
+corresponds to the parent r2 of r1. Since 1 + 5 = 6 we know that it is the leftmost child
+of the root. Hence the word ρ1ρ11ρ5 is obtained by reading the vertices of the leftmost
+subtree S of T1. We apply the ‘correction’ and we get ρ1ρ11ρ10 = Φ11(S). The rest of the
+word ρ1ρ11ρ10ρ2ρ11ρ11ρ9 corresponds to the other subtrees of T1 and it splits as ρ1ρ11ρ10
+and ρ2ρ11ρ11ρ9.
+Combining Proposition 3.8 and Proposition 3.10 we get our main result of the section:
+Theorem 3.12. For m = n − 1, the map Φm from the set of Schröder trees with n leaves
+to the set of words for ρn
+n−1 in Mn−1 is bijective.
+
+14
+THOMAS GOBET AND BAPTISTE ROGNERUD
+Corollary 3.13. The following two graphs are isomorphic under Φn−1:
+(1) The graph of words for ρn
+n−1 in Mn−1, where vertices are given by expressions of
+ρn
+n−1 and there is an edge between two expressions whenever they differ by applica-
+tion of a single relation,
+(2) The graph of Schröder trees with n leaves, where vertices are given by Schröder trees
+and there is an edge between two trees whenever they differ by application of a local
+move.
+Proof. The previous theorem gives the bijection between the sets of vertices. The proof of
+Lemma 3.7 shows that whenever one can apply a local move, one can apply a relation on
+the corresponding words. The proof of Proposition 3.8 shows that whenever one can apply
+a relation on words, a local move can be applied on the corresponding trees.
+□
+We illustrate the situation for M3 in Figure 6 below.
+Corollary 3.14. The number of words for the Garside element of Mn is a little Schröder
+number A001003 [11].
+Lemma 3.15. Let T = (r, S1, · · · , Sk) be a Schröder tree with n leaves labelled by m = n−1.
+Then, the word obtained by reading all the labels of a subtree Sj is a word for ρljn where lj
+is the number of leaves of Sj.
+Proof. Let us assume that the tree Sj has s + 1 leaves. By Lemma 3.7, the labels of the
+subtree Sj is a word for ρsρs
+n−1ρn−1−s. If s = 0, then we have a word for ρn−1. Otherwise,
+we can apply the relations [8, Lemma 4.5] with i = s and j = n−1. Alternatively, using the
+Schröder trees, it is easy to see that these relations comes from the following modifications
+of the trees.
+The word ρsρs
+n−1ρn−1−s correspond to the case where the tree Sj is the
+Schröder bush with s + 1 leaves. Using our local moves, we can modify it to the left comb.
+The corresponding word is now (ρ1ρs)sρn−1−s. Now we can inductively apply the local
+move to contract the edge between the root of T and the root left comb. The result is s+1
+empty trees attached to the root of T and the corresponding word is ρs+1
+n−1.
+□
+Proposition 3.16. Let n ≥ 1. We have the following isomorphisms of posets:
+(1) D0
+n ∼= Div(∆n−1), Dn+1
+n
+∼= Div(∆0) = {•},
+(2) For all 1 ≤ i ≤ n, Di
+n ∼= Div(∆n−i),
+where every set is ordered by the restriction of the left-divisibility order in the monoid Mk
+for suitable k.
+Proof. We begin by proving the second statement. An element x of Di
+n can be written in the
+form ρi
+nx′, where x′ is uniquely determined by cancellativity, and such that ρn is not a left-
+divisor of x′. In particular, there is y a divisor of ∆n such that ρi
+nx′y = ρn+1
+n
+, and y ̸= 1. We
+associate a tree (or rather a family of trees) to x as follows. Write x′ as a product a1a2 · · · aj
+of elements of S. Complete the word ρi
+na1a2 · · · aj to a word ρi
+na1a2 · · · ajb1b2 · · · bℓ for ∆n,
+i.e., choose a word b1b2 · · · bℓ for y.
+There are several possibilités for the bi’s, but the
+condition that x ∈ Di
+n ensures that, writing the corresponding Schröder tree in the form
+(r, T1, T2, . . . , Ti, S1, S2, · · · Sd), where the i first trees are empty trees with a single leaf,
+then a1a2 · · · aj has all its labels inside S1. Indeed, the labelling a1a2 · · · aj begins at the
+beginning (in the post-order convention) of the tree S1 since the trees T1, T2, . . . , Ti yield
+the label ρi
+n, and if another tree among S2, . . . , Sd was partly labelled by the ai’s, then a
+power of ρn would left-divide x′, since the word obtained from S1 is a power of ρn (lemma
+3.15). It is then possible to reduce all the trees S1, S2, . . . , Sd to a single tree S still having
+the labelling a1, a2, . . . , aj at the beginning, by first reducing S2, . . . , Sd to a set of empty
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+15
+ρ2ρ1ρ3ρ2ρ1ρ3
+ρ3ρ1ρ3ρ2ρ3
+ρ3ρ1ρ3ρ1ρ3ρ1
+ρ3ρ2ρ3ρ3ρ1
+ρ2ρ3ρ3ρ1ρ3
+ρ3ρ3ρ3ρ3
+ρ3ρ3ρ1ρ3ρ2
+ρ3ρ2ρ1ρ3ρ2ρ1
+ρ1ρ3ρ1ρ3ρ1ρ3
+ρ1ρ3ρ2ρ3ρ3
+ρ1ρ3ρ2ρ1ρ3ρ2
+1
+2
+2
+1
+3
+3
+3
+2
+1
+3
+3
+3
+1
+1
+1
+3
+3
+3
+1
+2
+3
+3
+1
+2
+3
+3
+3
+3
+3
+3
+3
+3
+3
+2
+1
+3
+3
+1
+2
+2
+1
+3
+1
+1
+1
+3
+3
+3
+2
+1
+3
+3
+3
+2
+1
+3
+2
+1
+3
+Figure 6. Illustration of Corollary 3.13 for n = 4: the graph of reduced words for
+∆3 and the isomorphic graph of Schröder trees on 4 = 3 + 1 leaves.
+trees �T2, . . . , �Td′ with single leafs, and then merging S1 and �T2 using a local move, then
+merging the resulting tree with �T3, and so on (see Figure 7 for an illustration). In this way
+we associate to x a Schröder tree of the form (r, T1, T2, . . . , Ti, S), where the Tk’s are empty
+trees with a single leaf, and the labelling corresponding to the chosen word a1a2 · · · aj is an
+initial section of the tree S (in fact, in algebraic terms, what we did is modify the word for
+y to get a suitable one yielding a unique tree after the empty trees). Note that by initial
+section we mean a prefix of the word obtained from the labelling of S read in post-order,
+where we exclude the label of the root, i.e., if the root has a label, then the prefix is strict.
+We denote by Sn,i the set of such Schröder trees, that is, those Schröder trees on n leaves
+with i + 1 child of the root, and such that the i first child are leafs. Note that the tree
+that we attached to x depends on a choice of word for x, but applying a defining relation
+
+16
+THOMAS GOBET AND BAPTISTE ROGNERUD
+in the word x corresponds to applying a local move in the tree S, and this cannot make S
+split into several trees since the root of S is frozen (its label corresponds to the last letter
+of y ̸= 1). Hence we can apply all local moves with all labels in the (strict) initial section
+corresponding to a word for x, and we keep a Schröder tree on n−i+1 leaves. In this way,
+forgetting the i first empty trees, what we attached to x is an equivalence class of a (strict)
+initial section of a Schröder tree on n − i + 1 leaves under local moves, that is, a divisor
+of ∆n−i. This mapping is injective since one can recover a word for x from the obtained
+Schröder tree on n − i + 1 leaves easily by mapping S to (r, T1, . . . , Ti, S), labelling such a
+tree, and reading the word obtained by reading the i first empty trees and then the initial
+section.
+It remains to show that it is surjective. Hence consider an initial section of a Schröder
+tree S on n − i leaves. We must show that, in the tree (r, T1, T2, . . . , Ti, S), the initial
+section of S is a word a1a2 · · · aj which labels an element x′ of D0
+n. Assume that ρn is a
+left-divisor of x′. Then, using local moves only involving those labels in the initial section
+of S corresponding to a word for x′, one can transform (r, T1, T2, . . . , Ti, S) into a tree of the
+form (r, T1, T2, . . . , Ti, Ti+1, S′
+1, . . . , S′
+e), i.e., S can be split into several trees, the first one
+(corresponding to ρn) being an empty tree. This is a contradiction: to split S into several
+trees, one would need to apply a local move involving the root of S, which is frozen since
+the initial section does not cover the root. Hence x′ ∈ D0
+n, and our mapping is surjective.
+This completes the proof of the second point, as it is clear that our mappings preserve
+left-divisibility.
+For the first point, we have Dn+1
+n
+= {ρn+1
+n
+}, hence there is nothing to prove. To show
+that D0
+n ∼= Div(∆n−1), one proceeds in a similar way as in the proof of point 1. Let x ∈ D0
+n
+and let y such that xy = ∆n. Choose words for x and y, and consider the corresponding
+Schröder tree T = (r, T1, . . . , Tk). Since x ∈ D0
+n, the initial section of T corresponding to
+the word for x must be a proper initial section of T1. Using local moves on T2, . . . , Tk (which
+amounts to changing the word for y), we can find a Schröder tree that is equivalent to T
+under local moves, and that is of the form (r, �T1, �T2), where �T1 still has the chosen word
+for x as a proper initial section, and �T2 is the empty tree with only one leaf. In particular
+�T1 is a Schröder tree on n leaf. Applying defining relations to words for x amounts to
+applying local moves inside the first tree, and arguing as in the first point this establishes
+the isomorphism of posets between D0
+n and Div(∆n−1).
+□
+r
+Ti
+S1
+�T1
+�T2
+−→
+r
+Ti
+r1
+S1
+�T1
+�T2
+−→
+r
+Ti
+r2
+r1
+S1
+�T1
+�T2
+=
+r
+Ti
+S
+Figure 7. Illustration for the proof of Proposition 3.16.
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+17
+4. Enumerative results
+We have already seen (Theorem 3.12) that the words for ρn+1
+n
+are in bijection with
+Schröder trees on n+1 leaves. In this section, we give some additional enumerative results
+for several families of particular elements of Mn.
+4.1. Number of simple elements.
+Corollary 4.1. Let n ≥ 2, and let An := |Div(∆n)|. Then
+An = 2A0 + 2An−1 +
+n−2
+�
+i=1
+Ai.
+(4.1)
+It follows that An = F2n, where F0, F1, F2, . . . denotes the Fibonacci sequence 1, 2, 3, 5, 8, ...
+inductively defined by F0 = 1, F1 = 2, and Fi = Fi−1 + Fi−2 for all i ≥ 2. The sequence of
+the Ans is referred as A001906 in [11].
+Proof. The equality (4.1) follows immediately from the disjoint union Div(∆n) = �
+0≤i≤n+1 Di
+n
+and Proposition 3.16. We have A0 = F0, A1 = 3 = F2, and it is elementary to check that
+the inductive formula given by 4.1 is also satisfied by the sequence F2n. This shows that
+An = F2n for all n ≥ 0.
+□
+Definition 4.2. We call the lattice (Div(∆n), ≤) the even Fibonacci lattice.
+4.2. Number of left-divisors of the lcm of the atoms and odd Fibonacci lat-
+tice. The set DivL(ρn
+n) of left-divisors of ρn
+n also forms a lattice under the restriction
+of left-divisibility, since it is an order ideal in the lattice (Div(∆n), ≤). In terms of the
+Garside monoid Mn, the element ρn
+n is both the left- and right-lcm of the generators
+S = {ρ1, ρ2, . . . , ρn} (see [8, Corollary 4.17]). For n ≥ 1 we set Bn := |DivL(ρn
+n)|.
+Lemma 4.3. We have Bn = F2n−1 for all n ≥ 1. The sequence of the Bns is referred as
+A001519 in [11].
+Proof. Let x ∈ Div(∆n). We claim that x ∈ DivL(ρn
+n) if and only if ρnx ∈ Div(∆n). Indeed,
+if x ≤ ρn
+n, there is y ∈ Mn such that xy = ρn
+n. We then have ρnxy = ρn+1
+n
+= ∆n, hence
+ρnx is a left-divisor of ∆n. Conversely, assume that ρnx ∈ Div(∆n). It follows that there
+is y ∈ Div(∆n) such that ρnxy = ∆n = ρn+1
+n
+. By cancellativity we get that xy = ρn
+n, hence
+x ∈ DivL(ρn
+n).
+It follows that DivL(ρn
+n) is in bijection with the set
+{ρnx | x ∈ Div(∆n)} ∩ Div(∆n).
+But this set is nothing but �
+1≤i≤n+1 Di
+n. It follows that
+Bn = |Div(∆n)| − |D0
+n| = |Div(∆n)| − |Div(∆n−1)|,
+where the last equality follows from point (1) of Proposition 3.16. By Corollary 4.1 we
+thus get that
+Bn = An − An−1 = F2n − F2n−2 = F2n−1,
+which concludes the proof.
+□
+Definition 4.4. We call the lattice (DivL(ρn
+n), ≤) the odd Fibonacci lattice.
+Both lattices for M3 are depicted in Figure 1.
+Remark 4.5. Note that the set of right-divisors of ρn
+n also has cardinality Bn: in fact, the
+two posets (DivL(ρn
+n), ≤L) and (DivR(ρn
+n), ≤R) are anti-isomorphic via x �→ x, where x is
+the element of Mn such that xx = ρn
+n (this element is unique by right-cancellativity).
+
+18
+THOMAS GOBET AND BAPTISTE ROGNERUD
+4.3. Number of words for the divisors of the Garside element.
+Lemma 4.6. Let T1 and T2 be two Schröder trees with n leaves labelled by m ≥ n − 1, and
+denote by m1 and m2 the corresponding words obtained by reading the labels in post-order.
+If the words m1 and m2 have a common prefix x1x2 · · · xl, then xi labels a leftmost child in
+T1 if and only if it labels a leftmost child in T2.
+Proof. We prove the result by induction on the number of leaves. If x1 · · · xl is obtained
+by reading all the vertices of T1 = (r, S1, · · · , Sk), then m1 = x1 · · · xl = m2 and by
+Proposition 3.10, we have T1 = T2, hence there is nothing to prove. Otherwise, let Sj be
+the first subtree of T1 which is not covered by the word x1 · · · xl, similarly let Uk the first
+subtree of T2 = (r, U1, · · · Uv) which is not covered by x1 · · · xl. Looking at the proof of
+Proposition 3.10, we see that the first subtrees S1, · · · , Sj−1 are completely determined by
+the word x1 · · · xl, hence we have j = k and Si = Ui for all i < k. Let xs be the letter of
+x1 · · · xl labelling the first vertex of Sj. Let m′
+1 be the subword of m1 and m′
+2 the subword
+of m2 starting at the xs. As explained in the proof of Proposition 3.10, we can determine
+the subword mj
+1 of m′
+1 which correspond to Sj. The trees Uj and Sj do not need to have the
+same number of leaves. If one of the trees, say Uj, has less leaves, then one can apply local
+moves in the trees Uj, Uj+1, · · · Uv as in the proof of Proposition 3.16 in order to obtain
+a tree ˜Uj with the same number of leaves as Sj. This will modify the word m′
+2, but not
+the prefix xs · · · xl, and xi labels a leftmost child in Uj if and only if it labels a leftmost
+child in ˜Uj (see Figure 7 for an illustration). After doing the modification, we consider the
+subword mj
+2 corresponding to the tree ˜Uj and apply the induction hypothesis to mj
+1 and
+mj
+2.
+□
+Theorem 4.7. The set of words for the left-divisors of ρn+1
+n
+is in bijection with the set of
+Schröder trees with n + 2 leaves.
+Proof. Let us denote by sk the number of Schröder trees with k + 1 leaves, and dk the
+number of words for the divisors of ρk+1
+k
+.
+Recall that Div(∆n) = �
+0≤i≤n+1 Di
+n, and let di
+n be the number of words for the elements
+of Di
+n. If i = n + 1, then ρn+1
+n
+is the only element of Di
+n and by Theorem 3.12, there are
+sn words for this element, hence we have dn+1
+n
+= sn.
+Let 0 ≤ i ≤ n and w = x1 · · · xl be a word for an element of Di
+n. The word w is a strict
+prefix of a Schröder tree T = (r, S1, · · · , Sk). By Lemma 2.5, w = w1w2 where w1 is a word
+for ρi
+n and ρn is not a left divisor of w2 (when i = 0 the word w1 is empty). Let Sj be the
+last subtree of T which has a vertex labelled by a letter of w1. We can apply a succession
+of defining relations to w1 in order to obtain ρi
+n. These relations correspond to local move
+in the trees S1, · · · , Sj which collapse all the trees S1, · · · Sj to empty trees. In order to
+reduce Sj to a list of empty trees we must use its root. Since the root is always the last
+label of the tree in post-order, the word w1 covers all the first j trees which have in total
+i leaves. Since ρn does not divide w2, we see that w2 is a (possibly empty) strict prefix
+of Sj+1. It is also possible to modify the trees Sj+2, · · · , Sk without changing the first j
+trees. Indeed, as in the proof of Proposition 3.16 we can reduce the trees Sj+2, · · · , Sk to
+empty trees and then merge them (until we can) to Sj+1.
+• When i = 0, after modification we obtain a tree ˜T = (r, ˜S, L) where L the empty
+tree, ˜S is a tree with n leaves and w = w2 is a strict prefix of ˜S.
+• When 1 ≤ i ≤ n, we obtain a tree ˜T = (r, S1, · · · , Sj, �Sj+1) and w2 is a strict prefix
+of the tree �Sj+1 with n + 1 − i leaves.
+In both cases, the tree �Sj+1 is obtained by possibly introducing new vertices to Sj+1,
+and as Figure 7 shows, these new vertices occur after the vertices of Sj+1, in post-order,
+
+ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
+19
+hence w2 is still a strict prefix of �Sj+1. Hence, we see that in the decomposition w = w1w2
+of Lemma 2.5, the word w1 is obtained by reading all the vertices of a Schröder tree with
+i leaves and w2 is a strict prefix of a Schröder tree, denoted by ˜S, with li + 1 leaves where
+li = n − i leaves if i ̸= 0 and li = n − 1 if i = 0.
+Let w = w1w2 be a word of an element of Di
+n with w2 having t letters. Let ˜S be a
+Schröder tree with li + 1 leaves having w2 as a strict prefix. Then, we construct a word
+γ(w) by first extracting ˜S, then labelling it accordingly to its number of leaves (i.e., with
+m = li) and finally taking its first t letters in post-order. Algebraically, it is easy to see
+how the word γ(w) is obtained from w2: if wi is the label of a leftmost child in ˜S, we have
+γ(w)i = wi. Otherwise, since the tree ˜Sj has li + 1 leaves, we have γ(w)i = wi − n + (li).
+A priori γ(w) depends on the choice of a tree ˜S, but Lemma 4.6 tells us that γ(w) only
+depends on w2. The word γ(w) is a prefix of a Schröder tree with li + 1 leaves, hence it
+is a word for a divisor of ∆li. We have obtained a map γ from the set of words for the
+elements of Di
+n to the set of words for the divisors of ∆li.
+Conversely, if z is a word of length k for a divisor of ∆li, it is a prefix (strict since the
+root is not contributing) of a Schröder tree S with li +1 leaves. We can view S as a subtree
+of a Schröder tree with n + 1 leaves by considering:
+• T = (r, S, L) when i = 0;
+• T = (r, δi, S) when i ≥ 1.
+Reading up to the first k letters of the subtree S produces a word w = w1w2 of an
+element of Di
+n such that γ(w) = z. Hence γ is surjective and we set ǫ(z) = w2. As before
+ǫ(z) only depends on z, not on the tree having z as a prefix.
+When i = 0, the map γ is injective, indeed if w and z are two words such that γ(w) =
+γ(z), then by Lemma 4.6 the labels of the leftmost child in γ(w) and γ(z) are the same,
+hence w and z are equal. This proves that d0
+n = dn−1.
+When i ≥ 1, then γ is far from being injective, since it forgets the first part of the tree.
+The set of words for the elements of Di
+n is the disjoint union of two sets E1 and E2 where
+E1 is the set of words w = w1w2 where w1 covers exactly one tree S1 and E2 is the set
+of words where w1 covers at least two trees. Note that when i = 1, the set E2 is empty
+otherwise both sets are non-empty. Indeed E2 contains at least all the words of the form
+ρi
+nw2 and E1 contains at least the words of the form ρi−1ρi−1
+n
+ρn−i−1w2 which correspond
+to the Schröder bush δi attached as the leftmost subtree of a Schröder tree.
+If z is a word for a divisor of ∆li, we compute the cardinality of the preimage of z by γ
+by looking at γ−1(z) ∩ E1 and γ−1(z) ∩ E2. If i = 1, we obviously only consider the first
+case. The elements of γ−1(z) ∩ E1 are obtained by concatenation of the word of a single
+Schröder tree with i leaves and ǫ(z), and the elements of γ−1(z) ∩ E2 are concatenation
+of the words of a forest with i leaves made of at least two Schröder tree and ǫ(z). Such a
+forest is nothing but a Schröder tree with i-leaves from which the root has been removed.
+So we have
+|γ−1(z) ∩ E1| = si−1 = |γ−1(z) ∩ E2|.
+Taking the sum on all possible words z, we have d1
+n = s0 · dn−1 and di
+n = 2 · si−1 · dn−i
+when n ≥ i ≥ 2.
+We have obtained:
+d0
+n = dn−1;
+d1
+n = s0 · dn−1 = dn−1;
+and
+di
+n = 2 · si−1 · dn−i when n ≥ i ≥ 2 and dn+1
+n
+= sn.
+
+20
+THOMAS GOBET AND BAPTISTE ROGNERUD
+By induction on the number of leaves, we have di = si+1, for every i ≤ n − 1, and
+dn = 2sn + 2
+n
+�
+i=2
+si−1sn−i+1 + sn
+= 3sn + 2
+n−1
+�
+i=1
+sisn−i.
+Using generating functions, it is not difficult to check that this implies that dn = sn+1, see
+for example [13, Theorem 5].
+□
+References
+[1] D. Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. 36 (2003), 647-683.
+[2] J. Birman, K.H. Ko, and S.J. Lee, A New Approach to the Word and Conjugacy Problems in the
+Braid Groups, Adv. in Math. 139 (1998), 322–353.
+[3] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271.
+[4] P. Dehornoy, F. Digne, D. Krammer, E. Godelle, and J. Michel. Foundations of Garside theory, Tracts
+in Mathematics 22, Europ. Math. Soc. (2015).
+[5] P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups,
+Proc. London Math. Soc. (3) 79 (1999), no. 3, 569-604.
+[6] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302.
+[7] F.A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. 20 (1969), no. 2,
+235–254.
+[8] T. Gobet, On some torus knot groups and submonoids of the braid groups, J. Algebra 607 (2022),
+Part B, 260-289.
+[9] T. Gobet, A new Garside structure on torus knot groups and some complex braid groups, preprint
+(2022), https://arxiv.org/abs/2209.02291.
+[10] J-L. Loday, Realization of the Stasheff polytope, Arch. Math., 83 (2004), 267-278.
+[11] OEIS Foundation Inc, The On-Line Encyclopedia of Integer Sequences, Published electronically at
+https://oeis.org
+[12] M. Picantin, Petits groupes gaussiens, PhD Thesis, Université de Caen, 2000.
+[13] F. Qi, B. Guo Some explicit and recursive formulas of the large and little Schröder numbers, Arab
+Journal of Mathematical Sciences Vol: 23, Issue: 2, Page: 141-147 (2017).
+Institut Denis Poisson, CNRS UMR 7350, Faculté des Sciences et Techniques, Université
+de Tours, Parc de Grandmont, 37200 TOURS, France
+Email address: thomas.gobet@lmpt.univ-tours.fr
+Institut de Mathématiques de Jussieu, Paris Rive Gauche (IMJ-PRG), Campus des Grands
+Moulins, Université de Paris - Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 PARIS
+Cedex 13, France
+Email address: baptiste.rognerud@imj-prg.fr
+

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+arXiv:2301.03970v1  [math.GR]  10 Jan 2023
+Ulam stability of lamplighters and Thompson groups
+Francesco Fournier-Facio and Bharatram Rangarajan
+January 11, 2023
+Abstract
+We show that a large family of groups is uniformly stable relative to unitary groups
+equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten
+p-norms. These include lamplighters Γ ≀ Λ where Λ is infinite and amenable, as well as
+several groups of dynamical origin such as the classical Thompson groups F, F ′, T and
+V . We prove this by means of vanishing results in asymptotic cohomology, a theory
+introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable
+for studying uniform stability.
+Along the way, we prove some foundational results
+in asymptotic cohomology, and use them to prove some hereditary features of Ulam
+stability. We further discuss metric approximation properties of such groups, taking
+values in unitary or symmetric groups.
+1
+Introduction
+Let Γ be a countable discrete group, and let U be a family of finite-dimensional unitary
+groups. The problem of stability asks whether every almost-homomorphism Γ → U ∈ U
+is close to a homomorphism. To formalize this we need to choose a norm, and a way to
+interpret these approximate notions. We focus on the classical setting of uniform defects and
+distances, with respect to submultiplicative norms.
+Let U := {(U(k), ∥·∥)} be a family of finite-dimensional unitary groups equipped with bi-
+invariant submultiplicative norms ∥·∥ (we allow U(k) to appear multiple times with different
+norms). For instance ∥ · ∥ could be the operator norm - the most classical case - or more
+generally a Schatten p-norm. Given a map φ : Γ → U(k), we define its defect to be
+def(φ) := sup
+g,h∈Γ
+∥φ(gh) − φ(g)φ(h)∥.
+Given another map ψ : Γ → U(k), we define the distance between them to be
+dist(φ, ψ) := sup
+g∈Γ
+∥φ(g) − ψ(g)∥.
+Definition 1.1. A uniform asymptotic homomorphism is a sequence of maps φn : Γ → U(kn)
+such that def(φn) → 0. We denote this simply by φ : Γ → U. We say that φ, ψ : Γ → U are
+uniformly asymptotically close if they have the same range degrees and dist(φn, ψn) → 0.
+The group Γ is uniformly U-stable if every uniform asymptotic homomorphism is uni-
+formly asymptotically close to a sequence of homomorphisms.
+1
+
+We can also talk quantitatively about stability, by asking how close a homomorphism we
+can choose, in terms of the defect. This leads to the notion of stability with a linear estimate,
+which will be relevant for us and which we define precisely in Section 2.1.
+Early mentions of similar problems can be found in the works of von Neumann [vN29]
+and Turing [Tur38]. In [Ula60, Chapter 6] Ulam discussed more general versions of stability,
+which has since inspired a large body of work. Uniform U-stability has been studied mostly
+when U is the family of unitary groups equipped with the operator norm, for which the
+notion is typically referred to as Ulam stability. In this contest, Kazhdan proved stability of
+amenable groups [Kaz82], while Burger, Ozawa and Thom proved stability of certain special
+linear groups over S-integers, and instability of groups admitting non-trivial quasimorphisms
+[BOT13].
+More recently, the second author, Glebsky, Lubotzky and Monod proved Ulam stabil-
+ity of certain lattices in higher rank Lie groups, with respect to arbitrary submultiplicative
+norms [GLMR23]. For the proof, they introduce a new cohomology theory, called asymptotic
+cohomology, and prove that stability is implied by the vanishing of certain asymptotic co-
+homology classes α ∈ H2
+a(Γ, V). We refer the reader to Section 2.2 for the relevant definitions.
+The goal of this paper is to further the understanding of asymptotic cohomology, and
+apply this to prove new stability results.
+The main one is the stability of the classical
+Thompson groups:
+Theorem 1.2 (Section 5). Thompson’s groups F, F ′, T and V are uniformly U-stable, with
+a linear estimate.
+As remarked by Arzhantseva and P˘aunescu [AP15, Open problem], the analogous state-
+ment for pointwise stability in permutation of F would imply that F is not sofic, thus proving
+at once the existence of a non-sofic group and the non-amenability of F: two of the most
+remarkable open problems in modern group theory. We will discuss these problems and their
+relation to our results in Section 7.
+Theorem 1.2 for F and F ′ will follow from a stability result for certain lamplighters.
+Given groups Γ, Λ, the corresponding lamplighter (or restricted wreath product) is the group
+Γ ≀ Λ = (⊕ΛΓ) ⋊ Λ, where Λ acts by shifting the coordinates.
+Theorem 1.3. Let Γ, Λ be two countable groups, where Λ is infinite and amenable. Then
+Γ ≀ Λ is uniformly U-stable, with a linear estimate.
+By itself, Theorem 1.3 provides a plethora of examples of uniformly U-stable groups, to a
+degree of flexibility that was not previously available. For instance, using classical embedding
+results [HNN49] it immediately implies the following:
+Corollary 1.4. Every countable group embeds into a 3-generated group which is uniformly
+U-stable, with a linear estimate.
+In particular, this gives a proof that there exist uncountably many finitely generated
+uniformly U-stable groups, a fact which could also be obtained by applying Kazhdan’s The-
+orem [Kaz82] to an infinite family of finitely generated amenable groups, such as the one
+constructed by B. H. Neumann [Neu37].
+2
+
+In order to obtain stability of F and F ′ from Theorem 1.3, we exploit coamenability.
+Recall that a subgroup Λ ≤ Γ is coamenable if the coset space Γ/Λ admits a Γ-invariant
+mean. It is well known that F ′ and F contain a coamenable lamplighter F ≀ Z. Therefore the
+stability of F and F ′ (Corollary 5.8) follows from Theorem 1.3, and the following result:
+Proposition 1.5. Let Λ ≤ Γ be coamenable. If Λ is uniformly U-stable with a linear estimate,
+then so is Γ.
+This can be seen as a relative version of the celebrated result of Kazhdan, stating that
+amenable groups are uniformly U-stable [Kaz82]. To complete the picture, we also prove
+another relative version of Kazhdan’s Theorem, which is sort of dual to Proposition 1.5:
+Proposition 1.6. Let N ≤ Γ be an amenable normal subgroup. If Γ is uniformly U-stable
+with a linear estimate, then so is Γ/N.
+The fact that Theorem 1.2 follows from Theorem 1.3 and Proposition 1.5 is not special
+to Thompson’s group F: this phenomenon is typical of several groups of piecewise linear and
+piecewise projective homeomorphisms, which enjoy some kind of self-similarity properties
+(Theorem 5.1 and Corollary 5.2). Stability of T and V then follow from these results, to-
+gether with a bounded generation argument analogous to the one from [BOT13] (Corollaries
+5.11 and 5.12).
+As we mentioned above, the tool underlying the proofs of Theorem 1.3 and Proposition
+1.5 is asymptotic cohomology, in particular the vanishing of certain classes in degree 2. In
+this framework, Theorem 1.3 takes the following form:
+Theorem 1.7. Let Γ, Λ be two countable groups, where Λ is infinite and amenable. Then
+Hn
+a(Γ ≀ Λ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V.
+Here the word finitary refers to the fact that these modules arise from stability problems
+with respect to finite-dimensional unitary representations. This hypothesis is crucial: see
+Remark 6.1. Propositions 1.5 and 1.6 also follow from results in asymptotic cohomology,
+that this time does not need the finitary assumption:
+Proposition 1.8. Let Λ ≤ Γ be coamenable. Then the restriction map Hn
+a(Γ, V) → Hn
+a(Λ, V)
+is injective, for all n ≥ 0 and all dual asymptotic Banach ∗Γ-modules V.
+Proposition 1.9. Let N ≤ Γ be an amenable normal subgroup. Then the pullback map
+Hn
+a(Γ/N, V) → Hn
+a(Γ, V) is an isomorphism, for all n ≥ 0 and all dual asymptotic Banach
+∗(Γ/N)-modules V.
+Despite the lack of a general theorem connecting the two theories, asymptotic cohomology
+seems to be closely connected to bounded cohomology, a well-established cohomology theory
+[Joh72, Gro82, Iva85, Mon01, Fri17] that has become a fundamental tool in rigidity theory.
+The vanishing result for asymptotic cohomology of lattices leading to stability [GLMR23]
+follows closely the vanishing result for bounded cohomology of high-rank lattices [BM99,
+BM02, MS04]. Similarly, our proofs of Theorem 1.7 and Propositions 1.8 and 1.9 follow
+closely the corresponding bounded-cohomological results: for Theorem 1.7 this was recently
+proven by Monod [Mon22], while for Proposition 1.8 this is a foundational result in bounded
+cohomology [Mon01, 8.6] (see also [MP03]), and Proposition 1.9 is an analogue of Gromov’s
+3
+
+Mapping Theorem [Gro82]. Note that the bounded cohomology of T and V has also been
+recently computed [FFLM21, MN21, And22], but only with trivial real coefficients, and our
+proofs are of a different nature.
+We thus hope that the steps we undertake to prove our main results will be useful to
+produce more computations in asymptotic cohomology, and therefore more examples of uni-
+formly U-stable, and in particular Ulam stable, groups.
+Our results have applications to the study of approximating properties of groups. While
+questions on pointwise approximation, such as soficity, hyperlinearity, and matricial finite-
+ness, are in some sense disjoint from the content of this paper, our stability results imply
+that some of the groups considered in this paper are not uniformly approximable with respect
+to the relevant families U (Corollary 7.6). We are also able to treat the case of symmetric
+groups endowed with the Hamming distance, by a more direct argument (Proposition 7.7).
+We end this introduction by proposing a question. There is a notion of strong Ulam stabil-
+ity, where the approximations take values in unitary groups of possibly infinite-dimensional
+Hilbert spaces, with the operator norm. It is a well-known open question whether strong
+Ulam stability coincides with amenability. In this direction it is known that strong Ulam
+stable groups have no non-abelian free subgroups [BOT13, Theorem 1.2], but there exist
+groups without non-abelian free subgroups that are not strong Ulam stable [Alp20].
+On the other hand, our results also prove uniform U-stability stability of the piecewise
+projective groups of Monod [Mon13] and Lodha–Moore [LM16], which are nonamenable and
+without free subgroups (see Section 5.2). Therefore we ask the following:
+Question 1.10. Let Γ be a countable group without non-abelian free subgroups.
+Is Γ
+uniformly U-stable (with a linear estimate)? Or at least Ulam stable?
+In particular, are all countable torsion groups Ulam stable?
+In other words: if Γ is not Ulam stable, must Γ contain a non-abelian free subgroup? To
+our knowledge it is not every known if groups admitting non-trivial quasimorphisms must
+contain non-abelian free subgroups: see [Man05] and [Cal10] for partial results in this direc-
+tion.
+Conventions: All groups are assumed to be discrete and countable. The set of natu-
+ral numbers N starts at 0. A non-principal ultrafilter ω on N is fixed for the rest of the paper.
+Outline: We start in Section 2 by reviewing the framework of asymptotic cohomology
+and its applications to stability, as developed in [GLMR23]. In Section 3 we discuss hered-
+itary properties for Ulam stability, and prove Propositions 1.5 and 1.6. We then move to
+lamplighters and prove Theorem 1.3 in Section 4, then to Thompson groups proving Theorem
+1.2 in Section 5. In Section 6 we provide examples showing that some of our results and some
+of the results from [GLMR23] are sharp, and conclude in Section 7 by discussing applications
+to the study of metric approximations of groups.
+Acknowledgements: The authors are indebted to Alon Dogon, Lev Glebsky, Alexander
+Lubotzky and Nicolas Monod for useful conversations.
+4
+
+2
+Uniform stability and asymptotic cohomology
+In this section, we shall briefly summarize the notion of defect diminishing that allows us
+to formulate the stability problem as a problem of lifting of homomorphisms with abelian
+kernel, which in turn motivates the connection to second cohomology. For a more detailed
+description, refer to Section 2 in [GLMR23].
+2.1
+Uniform stability and defect diminishing
+We begin by reviewing some basic notions of ultraproducts and non-standard analysis, be-
+fore formulating the stability problem as a homomorphism lifting problem. For this, it is
+convenient to describe a uniform asymptotic homomorphism (which is a sequence of maps)
+as one map of ultraproducts. This in turn allows us to perform a soft analysis to obtain
+a (true) homomorphism to a quotient group. Recall that ω is a fixed non-principal ultra-
+filter on N. The algebraic ultraproduct �
+ω Xn of an indexed collection {Xn}n∈N of sets is
+defined to be �
+ω Xn := �
+n∈N Xn/ ∼ where for {xn}n∈N, {yn}n∈N ∈ �
+n∈N Xn, we define
+{xn}n∈N ∼ {yn}n∈N if {n : xn = yn} ∈ ω. Ultraproducts can be made to inherit algebraic
+structures of their building blocks. For instance, for a group Γ, the ultraproduct �
+ω Γ, called
+the ultrapower and denoted ∗Γ, is itself a group. Another important example we will use is
+the field of hyperreals ∗R, the ultrapower of R.
+Objects (sets, functions, etc.) that arise as ultraproducts of standard objects are referred
+to as internal. Important examples of non-internal objects are the subsets ∗Rb of bounded
+hyperreals, consisting of elements {xn}ω ∈ ∗R for which there exists S ∈ ω and C ∈ R≥0 such
+that |xn| ≤ C for every n ∈ S, and the subset ∗Rinf of infinitesimals, consisting of elements
+{xn}ω ∈ ∗R such that for every real ε > 0, there exists S ∈ ω such that |xn| < ε for every
+n ∈ S.
+For x, y ∈ ∗R, write x = Oω(y) if x/y ∈ ∗Rb, and write x = oω(y) if x/y ∈ ∗Rinf. In
+particular, x ∈ ∗Rb is equivalent to x = Oω(1) while ε ∈ ∗Rinf is equivalent to ε = oω(1). The
+subset ∗Rb forms a valuation ring with ∗Rinf being the unique maximal ideal, with quotient
+∗Rb/∗Rinf ∼= R. The quotient map st : ∗Rb → R is known as the standard part map or limit
+along the ultrafilter ω. The previous construction can also be replicated for Banach spaces.
+Let {Wn}n∈N be a family of Banach spaces. Then W = �
+ω Wn can be given the structure
+of a ∗R-vector space. In fact, it also comes equipped with a ∗R-valued norm, allowing us to
+define the external subsets Wb and Winf. The quotient ˜
+W := Wb/Winf is a real Banach space.
+Given a uniform asymptotic homomorphism {φn : Γ → U(kn)}n∈N with def(φn) =: εn →
+0, construct the internal map φ : ∗Γ → �
+ω U(kn) where φ := �
+ω φn, with (hyperreal) defect
+ε := {εn}ω ∈ ∗Rinf. Then the question of uniform stability with a linear estimate can be
+rephrased as asking whether there exists an internal homomorphism ψ : ∗Γ → �
+ω U(kn) such
+that their (hyperreal) distance satisfies dist(φ, ψ) := {dist(φn, ψn)}ω = Oω(ε).
+The advantage of rephrasing the question in terms of internal maps is that an internal
+map φ : ∗Γ → �
+ω U(kn) with defect ε ∈ ∗Rinf induces a true homomorphism ˜φ : ∗Γ →
+�
+ω U(kn)/B(ε) where B(ε) is the (external) normal subgroup of �
+ω U(kn) comprising ele-
+ments that are at a distance Oω(ε) from the identity. In particular, the question of uniform
+stability with a linear estimate can equivalently be rephrased as asking whether given such
+an internal map φ, can the homomorphism ˜φ : ∗Γ → �
+ω U(kn)/B(ε) be lifted to an internal
+5
+
+homomorphism ψ : ∗Γ → �
+ω U(kn).
+Reinterpreting uniform stability with a linear estimate as a homomorphism lifting problem
+motivates a cohomological approach to capturing the obstruction. However, the obstacle here
+is that the kernel B(ε) of the lifting problem is not abelian. This can be handled by lifting
+in smaller steps so that each step involves an abelian kernel. Define a normal subgroup I(ε)
+of B(ε) comprising elements that are at a distance of oω(ε) from the identity. Then we can
+attempt to lift ˜φ : ∗Γ → �
+ω U(kn)/B(ε) to an internal map ψ : ∗Γ → �
+ω U(kn) that is a
+homomorphism modulo I(ε). The problem is simpler from the cohomological point of view:
+since the norms are submultiplicative, the kernel B(ε)/I(ε) of this lifting problem is abelian.
+The group Γ is said to have the defect diminishing property with respect to U if such a lift
+exists; more explicitly, Γ has the defect diminishing property if for every uniform asymptotic
+homomorphism φ : Γ → U there exists a uniform asymptotic homomorphism ψ with the
+same range such that dist(φ, ψ) = Oω(def(φ)) and def(ψ) = oω(def(φ)).
+Theorem 2.1 ([GLMR23, Theorem 2.3.11]). Γ has the defect diminishing property with
+respect to U if and only if Γ is uniformly U-stable with a linear estimate.
+The obstruction to such a homomorphism lifting, with an abelian kernel B(ε)/I(ε), can
+be carefully modeled using a cohomology H•
+a(Γ, W) so that H2
+a(Γ, W) = 0 implies the defect
+diminishing property (and consequently, uniform stability with a linear estimate).
+Here
+W = �
+ω u(kn) is an internal Lie algebra of �
+ω U(kn) equipped with an asymptotic action of
+the ultrapower ∗Γ constructed from the uniform asymptotic homomorphism φ that we start
+out with. The logarithm of the defect map
+∗Γ × ∗Γ →
+�
+ω
+U(kn) : (g1, g2) �→ φ(g1)φ(g2)φ(g1g2)−1
+would correspond to an asymptotic 2-cocycle in H2
+a(Γ, W). Such a cocycle is a coboundary
+in this setting (that is, it represents the zero class in H2
+a(Γ, W)), if and only if the defect
+diminishing property holds for the asymptotic homomorphism φ.
+2.2
+Asymptotic cohomology
+The reduction to a lifting problem with abelian kernel motivates a cohomology theory of Γ
+with coefficients in the internal Lie algebra W = �
+ω u(kn) of �
+ω U(kn), equipped with an
+asymptotic conjugation action of Γ. In this section we review the formal definition of this
+cohomology, and state some results from [GLMR23] that we shall need to work with it.
+Let (Vn)n≥1 be a sequence of Banach spaces, and let V := �
+ω
+Vn be their algebraic ultra-
+product: we refer to such V as an internal Banach space. For v ∈ V we denote by ∥v∥ the
+hyperreal (∥vn∥)ω ∈ ∗R. We then denote by
+Vb := {v ∈ V : ∥v∥ ∈ ∗Rb};
+Vinf := {v ∈ V : ∥v∥ ∈ ∗Rinf}.
+Then the quotient ˜V := Vb/Vinf is a real Banach space, whose norm is induced by the
+ultralimit of ∥ · ∥ on Vb. For each Vn denote by V #
+n its continuous dual, and let V# be the
+corresponding algebraic ultraproduct. The pairing ⟨·, ·⟩n : V #
+n × Vn → R induces a pairing
+V# × V → ∗R which descends to ˜V# × ˜V → R. We call V# the internal dual of V.
+6
+
+Now let Γ be a countable discrete group, and let π : ∗Γ×V → V be an internal map which
+preserves ∥ · ∥ and induces an isometric linear action ˜π : ∗Γ × ˜V → ˜V of ∗Γ. Such a map π
+is referred to as an asymptotic ∗Γ-action on V. We then call (π, V), or V, if π is understood
+from context, an asymptotic Banach ∗Γ-module. Given an internal Banach ∗Γ-module (π, V),
+the contragradient on each coordinate induces an internal map π# : ∗Γ × V# → V# mak-
+ing (π#, V#) into an asymptotic Banach ∗Γ-module. We call a module V a dual asymptotic
+Banach ∗Γ-module if V is the dual of some asymptotic ∗Γ-module denoted V♭. We decorate
+these definitions with the adjective finitary if each Vn is finite-dimensional.
+Now for each m ≥ 0 define the internal Banach space L∞((∗Γ)m, V) := �
+ω
+ℓ∞(Γm, Vn)
+(note that m is fixed and n runs through the natural numbers with respect to the ultrafilter
+ω). Similarly to before, for f ∈ L∞((∗Γ)m, V) we denote ∥f∥ := (∥fn∥)ω ∈ ∗R and
+L∞
+b ((∗Γ)m, V) := {f ∈ L∞((∗Γ)m, V) : ∥f∥ ∈ ∗Rb};
+L∞
+inf((∗Γ)m, V) := {f ∈ L∞((∗Γ)m, V) : ∥f∥ ∈ ∗Rinf}.
+Given an asymptotic ∗Γ-action π on V, we can construct a natural asymptotic ∗Γ-action
+ρm : ∗Γ × L∞((∗Γ)m, V) → L∞((∗Γ)m, V) given by
+(ρm(g)(f))(g1, g2, . . . , gm) := πG(g)f(g−1g1, . . . , g−1gm)
+(1)
+Then the quotient
+˜L∞((∗Γ)m, V) := L∞
+b ((∗Γ)m, V)/L∞
+inf((∗Γ)m, V)
+is again a real Banach space equipped with an isometric ∗Γ-action induced coordinate-wise
+by ρm, which defines the invariant subspaces ˜L∞((∗Γ)m, V)
+∗Γ.
+Now define the internal coboundary map
+dm : L∞((∗Γ)m, V) → L∞((∗Γ)m+1, V);
+dm(f)(g0, . . . , gm) :=
+m
+�
+j=0
+(−1)jf(g0, . . . , ˆgj, . . . , gm),
+(2)
+which descends to coboundary maps
+˜
+dm : ˜L∞((∗Γ)m, V) → ˜L∞((∗Γ)m+1, V).
+Since ˜
+dm is ∗Γ-equivariant, it defines the cochain complex:
+0
+˜
+d0
+−→ ˜L∞(∗Γ, V)
+∗Γ
+˜
+d1
+−→ ˜L∞((∗Γ)2, V)
+∗Γ
+˜d2
+−→ ˜L∞((∗Γ)3, V)
+∗Γ
+˜
+d3
+−→ · · ·
+Definition 2.2 ([GLMR23, Definition 4.2.2]). The m-th asymptotic cohomology of Γ with
+coefficients in V is
+Hm
+a (Γ, V) := ker(
+˜
+dm+1)/ im( ˜
+dm).
+7
+
+Other resolutions may also be used to compute asymptotic cohomology. Recall ([Mon01,
+5.3.2]) that a regular Γ-space S is said to be a Zimmer-amenable Γ-space if there exists a Γ-
+equivariant conditional expectation m : L∞(Γ×S) → L∞(S). Let S be a regular Γ-space with
+a Zimmer-amenable action of Γ, and let L∞((∗S)m, V) := �
+ω
+L∞
+w∗(Sm, Vn) (where L∞
+w∗(Sm, Vn)
+is the space of bounded weak-∗ measurable function classes from Sm to Vn). Again, the
+asymptotic ∗Γ-action on V gives rise to a natural asymptotic ∗Γ-action on L∞((∗Γ)m, V) as
+in (1), making L∞((∗S)m, V) an asymptotic Banach ∗Γ-module. The coboundary maps too
+can be defined just as in (2), to construct the cochain complex, and we have:
+Theorem 2.3 ([GLMR23, Theorem 4.3.3]). Let S be a Zimmer-amenable Γ-space, and V
+be a dual asymptotic Banach ∗Γ-module. Then H•
+a(Γ, V) can be computed as the asymptotic
+cohomology of the cochain complex
+0
+˜
+d0
+−→ ˜L∞(∗S, V)
+∗Γ
+˜
+d1
+−→ ˜L∞((∗S)2, V)
+∗Γ
+˜
+d2
+−→ ˜L∞((∗S)3, V)
+∗Γ
+˜
+d3
+−→ · · ·
+In the context of uniform U-stability, the relevant asymptotic Banach ∗Γ-module we shall
+be interested is the ultraproduct W = �
+ω u(kn), where u(kn) is the Lie algebra of U(kn). Note
+that we are only considering finite-dimensional unitary groups, so such a module is finitary.
+Given a uniform asymptotic homomorphism φ : ∗Γ → �
+ω U(n) with defect def(φ) ≤ω ε ∈
+∗Rinf, this can be used to construct the asymptotic action π : ∗Γ × W → W given by
+π(g)v = φ(g)vφ(g)−1, making W an asymptotic Banach ∗Γ-module. We call such a module
+an Ulam ∗Γ-module supported on U.
+Also, consider the map α : ∗Γ × ∗Γ → W given by
+α(g1, g2) = 1
+ε log(φ(g1)φ(g2)φ(g1g2)−1).
+(3)
+This map α induces an inhomogeneous 2-cocycle ˜α : ∗Γ × ∗Γ →
+˜
+W, and thus defines
+a class in H2
+a(Γ, W), under the usual correspondence between inhomogeneous cochains and
+invariant homogeneous cochains [GLMR23, Theorem 4.2.4]. We call such a class an Ulam
+class supported on U. This class vanishes, i.e. ˜α is a coboundary, precisely when φ has the
+defect diminishing property. Thus Theorem 2.1 yields:
+Theorem 2.4 ([GLMR23, Theorem 4.2.4]). Γ is uniformly U-stable with respect to U if and
+only if all Ulam classes supported on U vanish. In particular, if H2
+a(Γ, W) = 0 for every Ulam
+∗Γ-module supported on U, then Γ is uniformly U-stable, with a linear estimate.
+3
+Hereditary properties
+In this section, we first prove Proposition 1.8 and deduce Proposition 1.5 from it; then
+analogously we prove Proposition 1.9 and deduce Proposition 1.6 from it. Both stability
+statements are not symmetric, and in fact we will see in Section 6 that the converses do not
+hold.
+3.1
+More on Zimmer-amenability
+For the proofs of Propositions 1.8 and 1.9, we will need a more precise version of Theorem 2.3
+in a special case. A regular Γ-space S is said to be discrete if it is a countable set equipped
+8
+
+with the counting measure. It follows from the equivalent characterizations in [AEG94] that
+a discrete Γ-space is Zimmer-amenable precisely when each point stabilizer is amenable. In
+particular:
+1. If Λ ≤ Γ is a subgroup, then the action of Λ on Γ by left multiplication is free, so Γ is
+a discrete Zimmer-amenable Λ-space.
+2. If N ≤ Γ is an amenable subgroup, then the action of Γ on the coset space Γ/N has
+stabilizers equal to conjugates of N, so Γ/N is a discrete Zimmer-amenable Γ-space.
+For such spaces, we can provide an explicit chain map that implements the isomorphism
+in cohomology from Theorem 2.3. Indeed, the proof of Theorem 2.3 works by starting with
+a Γ-homotopy equivalence between the two complexes:
+0 → L∞(Γ) → L∞(Γ2) → L∞(Γ3) → · · ·
+0 → L∞(S) → L∞(S2) → L∞(S3) → · · ��
+which is then extended internally to the asymptotic version of these complexes. The case
+of dual asymptotic coefficients follows via some suitable identifications of the corresponding
+complexes (see the paragraph preceding [GLMR23, Theorem 4.20]). In the case of a discrete
+group Γ and a discrete Zimmer-amenable Γ-space, the homotopy equivalence above can be
+chosen to be the orbit map
+om
+b : L∞(Sm) −→ L∞(Γm)
+om
+b (f)(g1, . . . , gm) = f(g1b, . . . , gmb);
+where b ∈ S is some choice of basepoint [Fri17, Section 4.9]. Therefore in this case we obtain
+the following more explicit version of Theorem 2.3:
+Theorem 3.1. Let S be a discrete Zimmer-amenable Γ-space, with a basepoint b ∈ S, and
+let V be a dual asymptotic Banach ∗Γ-module. Then the orbit map
+om
+b : L∞((∗S)m, V) −→ L∞((∗Γ)m, V)
+om
+b (f)(g1, . . . , gm) = f(g1b, . . . , gmb);
+induces induces an isomorphism between H•
+a(Γ, V) and the cohomology of the complex
+0
+˜
+d0
+−→ ˜L∞(∗S, V)
+∗Γ
+˜
+d1
+−→ ˜L∞((∗S)2, V)
+∗Γ
+˜
+d2
+−→ ˜L∞((∗S)3, V)
+∗Γ
+˜
+d3
+−→ · · ·
+In the two basic examples of discrete Zimmer-amenable spaces from above, we obtain:
+Corollary 3.2. Let Λ ≤ Γ be a subgroup, and let V be a dual asymptotic Banach ∗Γ-
+module, which restricts to a dual asymptotic Banach ∗Λ-module.
+Then the restriction of
+cochains L∞((∗Γ)m, V) → L∞((∗Λ)m, V) induces an isomorphism between H•
+a(Λ, V) and the
+cohomology of the complex:
+0
+˜
+d0
+−→ ˜L∞(∗Γ, V)
+∗Λ
+˜
+d1
+−→ ˜L∞((∗Γ)2, V)
+∗Λ
+˜
+d2
+−→ ˜L∞((∗Γ)3, V)
+∗Λ
+˜
+d3
+−→ · · ·
+Proof. Seeing Γ as a discrete Zimmer-amenable Λ-space, with basepoint 1 ∈ Γ, the orbit map
+is nothing but the restriction of cochains, and we conclude by Theorem 3.1.
+9
+
+Corollary 3.3. Let N ≤ Γ be an amenable normal subgroup, and let V be a dual asymptotic
+Banach ∗(Γ/N)-module, which pulls back to a dual asymptotic Banach ∗Γ-module. Then the
+pullback of cochains L∞((∗(Γ/N))m, V) → L∞((∗Γ)m, V) induces an isomorphism between
+H•
+a(Γ, V) and H•
+a(Γ/N, V).
+Proof. Seeing Γ/N as a discrete Zimmer-amenable Γ-space, with basepoint the coset N, the
+orbit map is nothing but the pullback of cochains. So Theorem 3.1 yields an isomorphism
+between H•
+a(Γ, V) and the cohomology of the complex:
+0
+˜
+d0
+−→ ˜L∞(∗(Γ/N), V)
+∗Γ
+˜
+d1
+−→ ˜L∞((∗(Γ/N))2, V)
+∗Γ
+˜
+d2
+−→ ˜L∞((∗(Γ/N))3, V)
+∗Γ
+˜d3
+−→ · · ·
+But since the action of ∗Γ on both ∗(Γ/N) and V factors through ∗(Γ/N), the above complex
+coincides with the standard one computing H•
+a(Γ/N, V).
+We will use these explicit isomorphisms in this section. Later, for the proof of Theorem
+1.7, non-discrete Zimmer-amenable spaces will also appear, but in that case we will only need
+the existence of an abstract isomorphism as in Theorem 2.3.
+3.2
+Restrictions and coamenability
+Let Λ ≤ Γ be a (not necessarily coamenable) subgroup, and V be a dual asymptotic Ba-
+nach ∗Γ-module, which restricts to a dual asymptotic Banach ∗Λ-module. The restriction
+˜L∞((∗Γ)•, V)
+∗Γ → ˜L∞((∗Λ)•, V)
+∗Λ induces a map in cohomology, called the restriction map,
+and denoted
+res• : H•
+a(Γ, V) → H•
+a(Λ, V).
+This map behaves well with respect to Ulam classes:
+Lemma 3.4. Let W be an Ulam ∗Γ-module supported on U. Then W is also an Ulam ∗Λ-
+module supported on U, and the restriction map res2 : H2
+a(Γ, W) → H2
+a(Λ, W) sends Ulam
+classes to Ulam classes.
+Proof. Let φ : Γ → U be a uniform asymptotic homomorphism, and let W be the corre-
+sponding Ulam ∗Γ-module. Then restricting φn to Λ for each n yields a uniform asymptotic
+homomorphism φ|Λ : Λ → U, with def(φ|Λ) ≤ω def(φ) and endows W with an asymptotic
+∗Λ-action making it into an Ulam ∗Λ-module supported on U. The cocycle corresponding to
+φ is defined via the map
+α : ∗Γ × ∗Γ → W : (g1, g2) �→ 1
+ε log(φ(g1)φ(g2)φ(g1g2)−1).
+Since def(φ|Λ) ≤ω ε, restricting α to ∗Λ × ∗Λ yields a valid cocycle associated to the
+uniform asymptotic homomorphism φΛ. It follows that the chain map ˜L∞((∗Γ)•, V)
+∗Γ →
+˜L∞((∗Λ)•, V)
+∗Λ preserves the set of cocycles defined via uniform asymptotic homomorphisms,
+and therefore preserves Ulam classes.
+Now suppose that Λ ≤ Γ is coamenable. This means, by definition, that there exists a
+Γ-invariant mean on Γ/Λ; that is, there exists a linear functional m : ℓ∞(Γ/Λ) → R such
+that
+1. m(1Γ/Λ) = 1, where 1Γ/Λ denotes the constant function.
+10
+
+2. |m(f)| ≤ ∥f∥ for all f ∈ ℓ∞(Γ/Λ).
+3. m(g · f) = m(f) for all g ∈ Γ and all f ∈ ℓ∞(Γ/Λ).
+As with the absolute case [GLMR23, Lemma 3.20], we have the following:
+Lemma 3.5. Suppose that Λ ≤ Γ is coamenable, and let V be a dual asymptotic Banach
+∗Γ-module. Then there exists an internal map m : L∞(∗Γ/∗Λ, V) → V which induces a map
+˜m : ˜L∞(∗Γ/∗Λ, V) → ˜V with the following properties:
+1. If ˜f is the constant function equal to ˜v ∈ ˜V, then ˜m( ˜f) = ˜v.
+2. ∥ ˜m( ˜f)∥ ≤ ∥ ˜f∥ for all ˜f ∈ ˜L∞(∗Γ/∗Λ, V).
+3. ˜m(g · ˜f) = ˜m( ˜f) for all g ∈ ∗Γ and all ˜f ∈ ˜L∞(∗Γ/∗Λ, V).
+Proof. Consider f = {fn}ω ∈ L∞(∗Γ/∗Λ, V). Since V is a dual asymptotic ∗Γ-module with
+predual V♭, for each λ ∈ V♭, we get an internal map
+f λ : ∗Γ/∗Λ → ∗R : x �→ f(x)(λ).
+Note that f λ being internal, it is of the form {f λ
+n}ω where f λ
+n ∈ ℓ∞(Γ/Λ). This allows us to
+construct the internal map mλ
+in : L∞(∗Γ/∗Λ, V) → ∗R as
+mλ
+in(f) = {m
+�
+f λ
+n
+�
+}ω
+and finally min : L∞(∗Γ/∗Λ, V) → V as
+min(f)(λ) = mλ
+in(f)
+It is straightforward to check that min as defined induces a linear map ˜m : ˜L∞(∗Γ/∗Λ, V) → ˜V.
+As for ∗Γ-equivariance, this follows from the observation that (g · f)λ(x) = π(g)f(g−1x)(λ)
+while (g·f λ)(x) = f(g−1x)(λ). The conditions on ˜m follow from the definition and properties
+of the Γ-invariant mean m on ℓ∞(Γ/Λ).
+We are now ready to prove Proposition 1.8. The proof goes along the lines of [Mon01,
+Proposition 8.6.2].
+Proposition (Proposition 1.8). Let Λ ≤ Γ be coamenable. Then the restriction map Hn
+a(Γ, V) →
+Hn
+a(Λ, V) is injective, for all n ≥ 0 and all dual asymptotic Banach ∗Γ-modules V.
+Proof. We implement the asymptotic cohomology of Λ using the complex ˜L∞((∗Γ)•, V)
+∗Λ
+from Corollary 3.2. Since the chain map that defines the restriction map factors through
+this complex, and the chain map ˜L∞((∗Γ)•, V)
+∗Λ → ˜L∞((∗Λ)•, V)
+∗Λ induces an isomorphism
+in cohomology (Corollary 3.2), it suffices to show that the chain inclusion ˜L∞((∗Γ)•, V)
+∗Γ →
+˜L∞((∗Γ)•, V)
+∗Λ induces an injective map in cohomology. Henceforth, we will refer to this as
+the restriction map.
+Our goal is construct a transfer map, that is a linear map trans• : H•
+a(Λ, V) → H•
+a(Γ, V)
+such that trans• ◦ res• is the identity on H•
+a(Λ, V). Then it follows at once that res• must be
+injective. By the above paragraph, we may do this by constructing an internal chain map
+�
+trans
+• : ˜L∞((∗Γ)•, V)
+∗Λ → ˜L∞((∗Λ)•, V)
+∗Γ that restricts to the identity on ˜L∞((∗Λ)•, V)
+∗Γ.
+11
+
+Let f ∈ L∞((∗Γ)k, V) be such that ˜f ∈ ˜L∞((∗Γ)k, V)
+∗Λ. For each x ∈ (∗Γ)k, define
+fx : ∗Γ → V
+fx(g) := π(g)f(g−1x)
+In other words, fx(g) is just (ρ1(g)f)(x) as in (1). Since ˜f ∈ ˜L∞((∗Γ)k, V)
+∗Λ, for any γ ∈ ∗Λ
+and g ∈ ∗Γ,
+fx(gγ) − fx(g) ∈ Vinf
+Let us choose representatives of left ∗Λ-cosets in ∗Γ and restrict fx to this set of repre-
+sentatives so that we can regard fx as an internal map fx : ∗Γ/∗Λ → V. Moreover, since
+fx ∈ L∞(∗Γ/∗Λ, V), we can apply the mean m constructed in Lemma 3.5 to define the internal
+map transk(f) : L∞((∗Γ)k, V) → L∞((∗Γ)k, V) by
+transk(f)(x) = m(fx)
+Since ˜m is ∗Γ-invariant, this means that for g ∈ ∗Γ, m(fgx) − π(g)m(fx) ∈ Vinf, and implies
+that
+(transk(f))(gx) − π(g) transk(f)(x) ∈ Vinf.
+This establishes that for f ∈ L∞((∗Γ)k, V) with ˜f ∈ ˜L∞((∗Γ)k, V)
+∗Λ, we have
+�
+transk(f) ∈
+˜L∞((∗Γ)k, V)
+∗Γ. Therefore trans• induces a chain map
+˜
+trans
+• : ˜L∞((∗Γ)•, V)
+∗Λ → ˜L∞((∗Λ)•, V)
+∗Γ.
+Finally, if ˜f is already ∗Γ-invariant, then fx is constant up to infinitesimals, and thus m(fx)
+is equal, up to an infinitesimal, to the value of that constant, which is f(x). This shows that
+�
+trans
+k is the identity when restricted to ˜L∞((∗Λ)•, V)
+∗Γ, and concludes the proof.
+Proposition 1.5 is now an easy consequence.
+Proposition (Proposition 1.5). Let Λ ≤ Γ be coamenable. If Λ is uniformly U-stable with a
+linear estimate, then so is Γ.
+Proof. Suppose that Λ is uniformly U-stable with a linear estimate, and let Γ be a coamenable
+supergroup of Λ. We aim to show that Γ is also uniformly U-stable with a linear estimate.
+By Theorem 2.4, it suffices to show that all Ulam classes supported on U vanish in H2
+a(Γ, W),
+where W is an Ulam ∗Γ-module. Now by Proposition 1.8, it suffices to show that the images
+of such classes under the restriction map res2 : H2
+a(Γ, W) → H2
+a(Λ, W) vanish, since the latter
+is injective. By Lemma 3.4 these are Ulam classes of Λ. But since Λ is uniformly U-stable
+with a linear estimate, by Theorem 2.4 again, all Ulam classes in H2
+a(Λ, W) vanish, and we
+conclude.
+3.3
+Pullbacks and amenable kernels
+Let N ≤ Γ be an amenable normal subgroup, and let V be a dual asymptotic Banach ∗(Γ/N)-
+module, which pulls back to a dual asymptotic Banach ∗Γ-module. Precomposing cochains by
+the projection ∗Γ → ∗(Γ/N) defines the pullback p• : H•
+a(Γ/N, V) → H•
+a(Γ, V). The following
+can be proven via a similar argument as in Lemma 3.4:
+Lemma 3.6. Let W be an Ulam ∗(Γ/N)-module. Then W is also an Ulam ∗Γ-module, and
+the pullback p2 : H2
+a(Γ/N, W) → H2
+a(Γ, W) sends Ulam classes to Ulam classes.
+12
+
+With this language, Proposition 1.9 is just a reformulation of Corollary 3.3:
+Proposition (Proposition 1.9). Let N ≤ Γ be an amenable normal subgroup. Then the
+pullback Hn
+a(Γ/N, V) → Hn
+a(Γ, V) is an isomorphism, for all n ≥ 0 and all dual asymptotic
+Banach ∗Γ-modules V.
+And we deduce Proposition 1.6 analogously:
+Proposition (Proposition 1.6). Let N ≤ Γ be an amenable normal subgroup. If Γ is uni-
+formly U-stable with a linear estimate, then so is Γ/N.
+Proof. Suppose that Γ is uniformly U-stable with a linear estimate, and let N be an amenable
+normal subgroup of Γ. We aim to show that Γ/N is also uniformly U-stable with a linear
+estimate. By Theorem 2.4, it suffices to show that all Ulam classes supported on U vanish
+in H2
+a(Γ/N, W), where W is an Ulam ∗(Γ/N)-module. Now by Proposition 1.9, it suffices
+to show that the pullback of such classes under H2
+a(Γ/N, W) → H2
+a(Γ, W) vanish, since the
+latter is an isomorphism. By Lemma 3.6 these are Ulam classes of Γ. But since Γ is uniformly
+U-stable with a linear estimate, by Theorem 2.4 again, all Ulam classes in H2
+a(Γ, W) vanish,
+and we conclude.
+4
+Asymptotic cohomology of lamplighters
+In this section we prove Theorem 1.7, which we recall for the reader’s convenience:
+Theorem. Let Γ, Λ be two countable groups, where Λ is infinite and amenable.
+Then
+Hn
+a(Γ ≀ Λ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V.
+Remark 4.1. In fact, the theorem will hold for a larger class of coefficients, obtained as
+ultraproducts of separable Banach spaces. This does not however lead to a stronger stability
+result: see Remark 6.1.
+We start by finding a suitable Zimmer-amenable Γ-space:
+Lemma 4.2 ([Mon22, Corollary 8, Proposition 9]). Let Γ, Λ be two countable groups, where
+Λ is amenable. Let µ0 be a distribution of full support on Γ, and let µ be the product measure
+on S := ΓΛ. Then S is a Zimmer-amenable (Γ ≀ Λ)-space.
+The reason why this space is useful for computations is that it is highly ergodic. Recall that
+a Γ-space S is ergodic if every Γ-invariant function S → R is essentially constant. When S is
+doubly ergodic, that is the diagonal action of Γ on S ×S is ergodic, we even obtain ergodicity
+with separable coefficients, meaning that for every Γ-module E, every Γ-equivariant map
+S → E is essentially constant [Mon22, 2.A, 4.B].
+Lemma 4.3 (Kolmogorov [Mon22, 2.A, 4.B]). Let Γ, Λ be two countable groups, where Λ is
+infinite, and let S be as in Lemma 4.2. Then Sm is an ergodic (Γ≀Λ)-space, for every m ≥ 1.
+For our purposes, we will need an approximate version of ergodicity (namely, almost
+invariant functions are almost constant) and also the module E will only be endowed with
+an approximate action of Γ. The ergodicity assumption still suffices to obtain this:
+13
+
+Lemma 4.4. Let S be a probability measure Γ-space, and suppose that the action of Γ on S
+is ergodic. Then whenever f : S → R is a measurable function such that ∥g · f − f∥ < ε for
+all g ∈ Γ, there exists a constant c ∈ R such that |f(s) − c| < ε for almost every s ∈ S.
+Proof. We define F : S → R : s �→ esssupg∈Γf(g−1s). By construction, F is Γ-invariant,
+and moreover ∥F − f∥ < ε. By ergodicity, F is essentially equal to a constant c, and thus
+|f(s) − c| < ε for a.e. s ∈ S.
+Lemma 4.5. Let S be a probability measure Γ-space, and suppose that the action of Γ on
+S × S is ergodic. Suppose moreover, that E is a separable Banach space endowed with a map
+Γ × E → E : v �→ g · v such that ∥g · v∥ = ∥v∥ for all g ∈ Γ, v ∈ E.
+Then whenever f : S → E is a measurable function such that ∥g · f − f∥ < ε for all
+g ∈ Γ, where (g · f)(s) = g · f(g−1s), there exists a vector v ∈ E such that ∥f(s) − v∥ < 3ε
+for almost every s ∈ S.
+Proof. We define F : S × S → R : (s, t) �→ ∥f(s) − f(t)∥. Then
+∥g · F − F∥ = ess sup| ∥g · f(g−1s) − g · f(g−1t)∥ − ∥f(s) − f(t)∥ |
+≤ ess sup∥g · f(g−1s) − g · f(g−1t) − (f(s) − f(t))∥ ≤ 2∥g · f − f∥ < 2ε.
+By the previous lemma, there exists a constant c such that |F(s, t) − c| < 2ε for all ε > 0. If
+c < ε, then |f(s) − f(t)| < 3ε for a.e. s, t ∈ S, which implies the statement.
+Otherwise, ∥f(s) − f(t)∥ > ε for a.e. s, t ∈ S. Let D ⊂ E be a countable dense subset.
+Then for each d ∈ D the set f −1(Bε/2(d)) is a measurable subset of S, and the union of
+such sets covers S. Since D is countable, there must exist d ∈ D such that f −1(Bε/2(d)) has
+positive measure. But for all s, t in this set, ∥f(s) − f(t)∥ < ε, a contradiction.
+We thus obtain:
+Proposition 4.6. Let S be a doubly ergodic Γ-space. Let (Vn)n≥1 be a sequence of separable
+dual Banach spaces such that V = �
+ω
+Vn has the structure of a dual asymptotic Banach
+Γ-module be the corresponding asymptotic ∗Γ-module. Then the natural inclusion ˜V
+∗Γ →
+˜L∞(∗S, V)
+∗Γ is an isomorphism.
+Proof. Let f ∈ L∞
+b (∗S, V) = �
+ω
+L∞(S, Vn) be a lift of an element ˜f ∈ ˜L∞(∗S, V)
+∗Γ. We write
+f = (fn)ω. Then fact that ˜f is ∗Γ-invariant means that for every sequence (gn)n∈N ⊂ Γ it holds
+(gn·fn−fn)ω ∈ L∞
+inf(∗S, V). Since this holds for every sequence (gn)n∈N, a diagonal argument
+implies that there exists ε ∈ ∗Rinf such that for every g ∈ Γ it holds (g · fn −fn)ω ∈ ∗Rinf. It
+then follows from Lemma 4.5 that there exist (vn)ω ∈ V such that (fn − 1vn) ∈ L∞
+inf(∗S, V).
+Therefore f represents the same element of ˜L∞(∗S, V) as the image of an element of V. Since
+˜f is ∗Γ-invariant, the corresponding element is actually in ˜V
+∗Γ.
+We are finally ready to prove Theorem 1.7:
+Proof of Theorem 1.7. Let Γ, Λ be countable groups, where Λ is infinite and amenable. By
+Lemma 4.2, using the same notation, S is a Zimmer-amenable (Γ ≀ Λ)-space. Therefore we
+can apply Theorem 2.3, and obtain that the following complex computes H∗
+a(Γ ≀ Λ; V):
+0
+˜
+d0
+−→ ˜L∞(∗S, V)
+∗Γ
+˜
+d1
+−→ ˜L∞((∗S)2, V)
+∗Γ
+˜
+d2
+−→ ˜L∞((∗S)3, V)
+∗Γ
+˜
+d3
+−→ · · ·
+14
+
+Now by Lemma 4.3, Sm is a doubly ergodic (Γ ≀ Λ)-space, for every m ≥ 1. Thus Proposition
+4.6 applies, and the natural inclusion ˜V
+∗Γ → ˜L∞((∗S)m, V)
+∗Γ is an isomorphism for every
+m ≥ 1. Thus the above complex is isomorphic to
+0
+˜
+d0
+−→ ˜V
+∗Γ
+˜
+d1
+−→ ˜V
+∗Γ
+˜
+d2
+−→ ˜V
+∗Γ
+˜d3
+−→ · · ·
+Each differential ˜
+dm is an alternating sum of (m+1) terms all equal to each other. Therefore
+˜
+dm is the identity whenever m is even, and it vanishes whenever m is odd. The conclusion
+follows.
+5
+Thompson groups
+In this section we prove Theorem 1.2. The statement for F ′ will be a special case of a more
+general result for a large family of self-similar groups. The most general statement is the
+following:
+Theorem 5.1. Let Γ be a group, Γ0 a subgroup with the following properties:
+1. There exists g ∈ Γ such that the groups {giΓ0g−i : i ∈ Z} pairwise commute;
+2. Every finite subset of Γ is contained in some conjugate of Γ0.
+Then Hn
+a(Γ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V. In
+particular, Γ is uniformly U-stable, with a linear estimate.
+The theorem applies to the following large family of groups of homeomorphisms of the
+real line:
+Corollary 5.2. Let Γ be a proximal, boundedly supported group of orientation-preserving
+homeomorphisms of the line. Then Hn
+a(Γ, V) = 0 for all n ≥ 1 and all finitary dual asymptotic
+Banach ∗Γ-modules V. In particular, Γ is uniformly U-stable, with a linear estimate.
+Remark 5.3. The fact that such groups have no quasimorphisms is well-known: see e.g.
+[GG17, FFL21, Mon22].
+We refer the reader to Section 5.2 for the relevant definitions. In Corollary 5.8 we will
+apply Corollary 5.2 to Thompson’s group F ′; the result for Thompson’s group F will follow
+from Proposition 1.5. We deduce the stability of Thompson’s group T and V from these
+general criteria in Section 5.3.
+5.1
+Self-similar groups
+In this section we prove Theorem 5.1. This will be done in a series of lemmas:
+Lemma 5.4. Let Γ be a group, and suppose that there exists g ∈ Γ and Γ0 ≤ Γ such that
+{giΓ0g−i : i ∈ Z} pairwise commute. Then there exists an epimorphism Γ0 ≀ Z → ⟨Γ0, g⟩ with
+amenable (in fact, metabelian) kernel.
+This is well-known and stated without proof in [Mon22]. We include a proof for com-
+pleteness.
+15
+
+Proof. To make a clear distinction, we denote by H the abstract group Γ0, and by Γ0 the
+subgroup of Γ. So we want to construct an epimorphism H ≀Z → ⟨Γ0, g⟩ ≤ Γ with metabelian
+kernel. We define naturally
+ϕ((gi)i∈Z, p) =
+��
+i∈Z
+tigit−i
+�
+tp.
+Note that this product is well-defined since there are only finitely many non-identity terms,
+and the order does not matter since different conjugates commute. By construction ϕ is
+injective on Hi, that is the copy of H supported on the i-th coordinate in H ≀ Z.
+Let
+K := ker ϕ ∩ �
+i Hi, and note that K is the kernel of the retraction H ≀ Z → Z restricted to
+ker ϕ. So it suffices to show that K is abelian.
+Let g, h ∈ K and write them as (gi)i∈Z and (hi)i∈Z (we omit the Z-coordinate since it is
+always 0). We need to show that g and h commute. We have
+1Γ = ϕ(g) =
+�
+i∈Z
+tigit−i
+and thus
+g0 =
+�
+i̸=0
+tigit−i ∈ Γ.
+But now g0 belongs to a group generated by conjugates of Γ0 in Γ that commute with it. In
+particular this implies that g0 and h0 commute in Γ. Since ϕ|H0 is injective, this shows that
+g0 and h0 commute in H0. Running the same argument on the other coordinates, we obtain
+that gi and hi commute in Hi, for all i ∈ Z, and thus g and h commute.
+The next facts are all contained in the literature:
+Lemma 5.5 ([Mon22, Proposition 10]). Suppose that Γ0 ≤ Γ is such that every finite subset
+of Γ is contained in some Γ-conjugate of Γ0. Then Γ0 is coamenable in Γ.
+Lemma 5.6 ([MP03]). Let K ≤ H ≤ Γ.
+1. If K is coamenable in Γ, then H is coamenable in Γ;
+2. If K is coamenable in H and H is coamenable in Γ, then K is coamenable in Γ.
+Remark 5.7. We warn the reader that if K is coamenable in Γ, then K need not be
+coamenable in H [MP03].
+We are now ready to prove Theorem 5.1:
+Proof of Theorem 5.1. Let Γ, Γ0 and g be as in the statement. By Lemma 5.4, there exists
+a map Γ0 ≀ Z → ⟨Γ0, g⟩ with metabelian kernel. By Theorem 1.7 and Proposition 1.9, we
+have Hn
+a(⟨Γ0, g⟩, V) for all n ≥ 1 and all finitary dual asymptotic Banach ∗Γ-modules V. Now
+by Lemma 5.5, Γ0 is coamenable in Γ. Finally, by Lemma 5.6, ⟨Γ0, g⟩ is coameanble in Γ.
+Proposition 1.8 allows to conclude.
+5.2
+Groups of homeomorphisms of the line
+Let Γ be a group acting by homeomorphisms on the real line. We say that the action is
+proximal if for all reals a < b and c < d there exists g ∈ Γ such that g · a < c < d < g · b.
+The support of g ∈ Γ is the set {x ∈ R : g · x ̸= x}. We say that Γ is boundedly supported if
+every element has bounded support. Note that boundedly supported homeomorphisms are
+automatically orientation-preserving.
+16
+
+Proof of Corollary 5.2. Let Γ be as in the statement. Let Γ0 be the subgroup of elements
+whose support is contained in [0, 1]. Let g ∈ Γ be such that g(0) > 1: such an element exists
+because the action of Γ is proximal. Then it follows by induction, and the fact that Γ is
+orientation-preserving, that the intervals {gi[0, 1] : i ∈ Z} are pairwise disjoint. Therefore
+the conjugates giΓ0g−i pairwise commute.
+Since Γ is boundedly supported, for every finite subset A ⊂ Γ there exists n such that
+the support of each element of A is contained in [−n, n]. By proximality, there exists h ∈ Γ
+such that h(0) < −n and h(1) > n. Then hΓ0h−1 is the subgroup of elements whose support
+is contained in [−n, n], in particular it contains A.
+Thus Theorem 5.1 applies and we conclude.
+Let us now show how to obtain the statements on F and F ′ from Theorem 1.2 from
+Corollary 5.2 and Proposition 1.5.
+We refer the reader to [CFP96] for more details on
+Thompson’s groups.
+Thompson’s group F is the group of orientation-preserving piecewise linear homeomor-
+phisms of the interval, with breakpoints in Z[1/2] and slopes in 2Z. The derived subgroup F ′
+coincides with the subgroup of boundedly supported elements. The action of F ′ (and thus
+F) on [0, 1] preserves Z[1/2] ∩ (0, 1), and acts highly transitively on it; that is, for every pair
+of ordered n-tuples in Z[1/2] ∩ (0, 1) there exists an element of F ′ sending one to the other.
+Corollary 5.8. Thompson’s groups F and F ′ are uniformly U-stable, with a linear estimate.
+Proof. We identify (0, 1) with the real line. The group F ′ is boundedly supported, and it is
+proximal, since it acts transitively on ordered pairs of a dense set. Therefore Corollary 5.2
+applies and F ′ is uniformly U-stable, with a linear estimate.
+Since the quotient F/F ′ is abelian, thus amenable, we see that F ′ is coamenable in F, and
+thus conclude from Proposition 1.5 that F ′ is uniformly U-stable, with a linear estimate.
+Remark 5.9. We could also deduce the stability of F from the stability of F ′ more directly,
+without appealing to Proposition 1.5. Indeed, since F ′ is uniformly U-stable, simple, and
+not linear, every homomorphism F ′ → U(n) is trivial - something we will come back to in
+the next section. Therefore uniform U-stability of F ′ implies that every uniform asymptotic
+homomorphism F ′ → U is uniformly close to the trivial one. It follows that every uniform
+asymptotic homomorphism F → U is uniformly asymptotically close to one that factors
+through Z2. We conclude by the stability of amenable groups [Kaz82, GLMR23].
+Other groups to which these criteria apply include more piecewise linear groups [BS16],
+such as the Stein–Thompson groups [Ste92], or the golden ratio Thompson group of Cleary
+[Cle00, BNR21]. In such generality some more care is needed, since the commutator subgroup
+is sometimes a proper subgroup of the boundedly supported subgroup. The criteria also apply
+for the piecewise proejective groups of Monod [Mon13] and Lodha–Moore [LM16]. In this
+case, further care is needed, since the role of the commutator subgroup in the proofs above
+has to be taken by the double commutator subgroup [BLR18]. This ties back to Question
+1.10 from the introduction.
+5.3
+T and V
+In this section, we show how our previous results allow to prove stability of groups of home-
+omorphisms of the circle and of the Cantor set as well. For simplicity of the exposition, we
+17
+
+only focus on Thompson’s groups T and V , but the proofs generalize to some analogously
+defined groups, with the appropriate modifications. Our proof will involve a bounded gen-
+eration argument for stability that was pioneered in [BOT13]. We will only use it a simple
+version thereof, closer to the one from [BC20]. Recall that Γ is said to be boundedly generated
+by the collection of subgroups H if there exists k ≥ 1 such that the sets {H1 · · · Hk : Hi ∈ H}
+cover Γ.
+Lemma 5.10. Let Γ be a discrete group. Suppose that there exists a subgroup H ≤ Γ with
+the following properties:
+1. Every homomorphism H → U(n) is trivial;
+2. H is uniformly U-stable (with a linear estimate);
+3. Γ is boundedly generated by the conjugates of H.
+Then Γ is uniformly U-stable (with a linear estimate).
+Proof. Let φn : Γ → U(dn) be a uniform asymptotic homomorphism with def(φn) =: εn.
+Then φn|H : H → U(dn) is a uniform asymptotic homomorphism of H, therefore it is δn-
+close to a homomorphism, where δn → 0. But by assumption such a homomorphism must
+be trivial, so ∥φn(h) − Ikn∥ ≤ δn for all n. The same holds for all conjugates of H, up to
+replacing δn by δn + 2εn.
+By bounded generation, there exists k ≥ 1 such that each g ∈ Γ can be written as
+g = h1 · · · hk, where each hi belongs to a conjugate of H. We estimate:
+∥φn(g) − Idn∥ =
+�����φn
+� k
+�
+i=1
+hi
+�
+− Idn
+����� ≤
+�����φn
+�k−1
+�
+i=1
+hi
+�
+φn(hk) − Idn
+����� + εn
+=
+�����φn
+�k−1
+�
+i=1
+hi
+�
+− Idn
+����� + ∥φn(hk) − Idn∥ + εn ≤ · · ·
+· · · ≤
+k
+�
+i=1
+∥φn(hi) − Idn∥ + kεn ≤ k(δn + εn).
+Therefore φn is k(δn + εn)-close to the trivial homomorphism, and we conclude.
+Thompson’s group T is the group of orientation-preserving piecewise linear homeomor-
+phisms of the circle R/Z preserving Z[1/2]/Z, with breakpoints in Z[1/2]/Z, and slopes in
+2Z. Given x ∈ Z[1/2]/Z, the stabilizer of x is naturally isomorphic to F. Moreover, the germ
+stabilizer T(x) (i.e. the group consisting of elements that fix pointwise some neighbourhood
+of x) is isomorphic to F ′.
+Corollary 5.11. Thompson’s group T is uniformly U-stable with a linear estimate.
+Proof. We claim that Lemma 5.10 applies with H = T(0) ∼= F ′. Item 1. follows from the
+fact F ′ does not embed into U(n) (for instance because it contains F as a subgroup, which
+is finitely generated and not residually finite, and so cannot be linear by Mal’cev’s Theorem
+[Mal40]), and F ′ is simple [CFP96]. Also, F ′ is uniformly U-stable with a linear estimate,
+by Corollary 5.8. Therefore we are left to show the bounded generation statement. We will
+18
+
+show that for every g ∈ T there exist x, y ∈ Z[1/2]/Z such that g ∈ T(x)T(y). This suffices
+because T acts transitively on Z[1/2]/Z, so T(x) and T(y) are both conjugate to H = T(0).
+Let 1 ̸= g ∈ T, and choose x ̸= y ∈ Z[1/2]/Z such that g(y) /∈ {x, y}. Let I be a small
+dyadic arc around y such that x /∈ I and x, y /∈ g(I). Choose an element f ∈ T(x) such
+that f(I) = g(I). Let h be an element supported on I such that h|I = f −1g|I. Since x /∈ I,
+we also have h ∈ T(x). Moreover h−1f −1g|I = id|I, so h−1f −1g ∈ G(y). We conclude that
+g = fh · h−1f −1g ∈ T(x)T(y).
+Thompson’s group V can be described as a group of homeomorphisms of the dyadic Cantor
+set X := 2N. A dyadic brick is a clopen subset of the form Xσ := σ × 2N>k, for some σ ∈ 2k,
+and every two dyadic bricks are canonically homeomorphic via Xσ → Xτ : σ × x �→ τ × x.
+An element g ∈ V is defined by two finite partitions of V of the same size into dyadic bricks,
+that are sent to each other via canonical homeomorphisms.
+Corollary 5.12. Thompson’s group V is uniformly U-stable, with a linear estimate.
+The proof is very similar to the proof for T, so we only sketch it:
+Sketch of proof. Let x ∈ 2N be a dyadic point, that is a sequence that is eventually all 0,
+and let V (x) denote the subgroup of V consisting of elements that fix a neighbourhood of x
+pointwise. The same argument as in the proof of Corollary 5.11 shows that V is boundedly
+generated by conjugates of V (x).
+Now V (x) is isomorphic to a directed union of copies of V , which is finitely generated
+and simple [CFP96], so by Mal’cev’s Theorem every homomorphism V (x) → U(n) is trivial.
+Finally, V (x) contains a copy V0 of V such that the pair (V (x), V0) satisfies the hypotheses of
+Theorem 5.1 (see [And22, Proposition 4.3.4] and its proof). We conclude by Lemma 5.10.
+6
+Sharpness of our results
+In this section we point out certain ways in which our results are sharp, by providing explicit
+counterexamples to generalizations and converses.
+Remark 6.1. There is a notion of strong Ulam stability, where one takes U to include unitary
+groups of infinite-dimensional Hilbert spaces as well, typically equipped with the operator
+norm. It is shown in [BOT13] that a subgroup of a strongly Ulam stable group is Ulam
+stable. Therefore it is clear that Theorem 1.3 does not hold for strong Ulam stability. Even
+restricting to separable Hilbert spaces does not help: it follows from the construction in
+[BOT13] that if a countable group contains a free subgroup, then separable Hilbert spaces
+already witness the failure of strong Ulam stability.
+The framework of stability via asymptotic cohomology can be developed in this general
+setting as well, with dual asymptotic Banach modules that are not finitary. Therefore the
+counterexample above shows that Theorem 1.7 really needs the finitary assumption. The fact
+that we could obtain dual asymptotic Banach modules obtained as ultraproducts of separable
+spaces, analogously to [Mon22], does not help, since the dual asymptotic Banach modules
+arising from a stability problem over infinite-dimensional Hilbert spaces are not of this form,
+even when the Hilbert space are separable.
+19
+
+Remark 6.2. We proved in Proposition 1.5 that if Λ is coamenable in Γ and Λ is uniformly
+U-stable with a linear estimate, then so is Γ. The converse does not hold. Let Fn be a free
+group of rank n ≥ 2. Then Λ := �
+n≥1 Fn admits a non-trivial quasimorphism, so it is not
+uniformly U(1)-stable [BOT13], in particular it is not uniformly U-stable. However, Λ is
+coamenable in Fn ≀ Z, which is uniformly U-stable with a linear estimate by Theorem 1.3.
+On the other hand, if we replace “coamenable” by “finite index”, then the converse does
+hold. This follows from the induction procedure in [BOT13] for Ulam stability, as detailed
+in [Gam11, Lemma II.22]; the same proof can be generalized to all submultiplicative norms
+[GLMR23, Lemma 4.3.6].
+Remark 6.3. We proved in Proposition 1.6 that if N is an amenable normal subgroup of Γ,
+and Γ is uniformly U-stable with a linear estimate, then so is Γ/N. The converse does not
+hold. Let Γ be the lift of Thompson’s group T, that is, the group of orientation-preserving
+homeomorphisms of R that commute with the group Z of integer translations and induce T
+on the quotient R/Z. These groups fit into a central extension
+1 → Z → Γ → T → 1.
+Now T is uniformly U-stable with a linear estimate, by Corollary 5.11, however Γ is not:
+it is not even uniformly U(1)-stable, by [BOT13], since it has a non-trivial quasimorphism
+[GS87].
+The next two remarks show that some results from [GLMR23] are also sharp.
+Remark 6.4. The fundamental result of [GLMR23] is that the vanishing of asymptotic
+cohomology implies uniform U-stability. The converse does not hold. Indeed, since u(1) ∼= R
+with trivial adjoint action (because U(1) is abelian), it follows that the implication of Theorem
+2.4 specializes to: If H2
+a(Γ, ∗R) = 0, then Γ is uniformly U(1)-stable, where ∗R is seen as a
+dual asymptotic ∗Γ-module with a trivial ∗Γ action.
+Now, let again Γ be the lift of Thompson’s group T, so that Γ contains a central subgroup
+Z with Γ/Z ∼= T. The fact that Γ is not uniformly U(1)-stable implies that H2
+a(Γ, ∗R) ̸= 0.
+But Proposition 1.9 then shows that H2
+a(T, ∗R) ̸= 0 either. However, T is uniformly U-stable
+with a linear estimate, by Corollary 5.11. Morally, this is due to the fact that H2
+b(Γ, R) ∼=
+H2
+b(T, R) ∼= R, but the former is spanned by a quasimorphisms, while the latter is not (see
+e.g. [Cal09, Chapter 5]).
+Remark 6.5. In [BOT13] it is shown that groups admitting non-trivial quasimorphisms are
+not uniformly U(1)-stable. In [GLMR23, Proposition 1.0.6] this result is sharpened: the
+authors show that Γ is uniformly U(1)-stable if and only if the non-zero element in the image
+of H2
+b(Γ, Z) in H2
+b(Γ, R) have Gromov norm ∥·∥ bounded away from 0. They use this to show
+that a finitely presented group is uniformly U(1)-stable if and only if it admits no non-trivial
+quasimorphism [GLMR23, Corollary 1.0.10].
+The hypothesis of finite presentability is necessary. Let Γn denote the lift of Thompson’s
+group T to R/nZ. That is, Γn is the group of orientation-preserving homeomorphisms of the
+topological circle R/nZ, which commute with the cyclic group of rotations Z/nZ and induce
+T on the quotient R/Z. Now T has no unbounded quasimorphisms (see e.g. [Cal09, Chapter
+5]), and so Γn also has no unbounded quasimorphisms (this follows from the left exactness
+of the quasimorphism functor [Cal09, Remark 2.90]). Therefore the group Γ := �
+n≥2 Γn has
+no unbounded quasimorphisms.
+20
+
+However, we claim that Γ is not uniformly U(1)-stable. By [GLMR23, Proposition 1.0.6],
+it suffices to show that there exist bounded cohomology classes 0 ̸= ρn ∈ im(H2
+b(Γ, Z) →
+H2
+b(Γ, R) such that ∥ρn∥ → 0. We let ρn be the Euler class of the representation Γ → Γn →
+Homeo+(R/nZ), which admits an integral representative and so lies in the image of H2
+b(Γ, Z)
+(see [Ghy01] for more information about Euler classes of circle actions). Moreover, using the
+terminology of [Bur11], the representation is minimal, unbounded, and has a centralizer of
+order n. Therefore ∥ρn∥ = 1/2n by [Bur11, Corollary 1.6], and we conclude.
+Note that Γ is countable but infinitely generated. It would be interesting to produce a
+finitely generated example (which would necessarily be infinitely presented).
+7
+Approximation properties
+In this section we discuss open problems about approximation properties of the groups treated
+in this paper, and their relation to our results. We recall the following notions:
+Definition 7.1. Let G be a family of metric groups. We say that Γ is (pointwise, uniformly)
+G-approximable if there exists a (pointwise, uniform) asymptotic homomorphism φn : Γ →
+Gn ∈ G that is moreover asymptotically injective, meaning that for all g ∈ Γ, g ̸= 1 it holds
+lim inf
+n→∞ φn(g) > 0.
+The above terminology is not standard: most of the literature only deals with the point-
+wise notion, and refers to that as G-approximability. The notion of uniform approximability
+appeared in [FF21] with the name of strong G-approximability.
+Example 7.2. If G is the family of symmetric groups equipped with the normalized Hamming
+distance, then pointwise G-approximable groups are called sofic [Gro99, Wei00].
+If G is the family of unitary groups equipped with the Hilbert–Schmidt distance, then
+pointwise G-approximable groups are called hyperlinear [R˘08].
+All amenable and residually finite groups are sofic, and all sofic groups are hyperlinear.
+It is a major open question to determine whether there exists a non-sofic group.
+In our context of submultiplicative norms on unitary groups, the following two notions of
+approximability have been studied:
+Example 7.3. Let G be the family of unitary groups equipped with the operator norm.
+Then pointwise G-approximable groups are called MF [CDE13]. All amenable groups are
+MF [TWW17]. It is an open problem to determine whether there exists a non-MF group.
+Let G be the family of unitary groups equipped with the Frobenius norm, or more generally
+with a Schatten p-norm, for 1 < p < ∞. Groups that are not pointwise G-approximable have
+been constructed in [DCGLT20, LO20]. This is one of the very few cases in which a non-
+example for pointwise approximability is known.
+The following observation is well-known, and due to Glebsky and Rivera [GR09] and
+Arzhantseva and P˘aunescu in the pointwise symmetric case [AP15]. We give a general proof
+for reference:
+Proposition 7.4. Let G be a family of metric groups that are locally residually finite, and
+let Γ be a finitely generated group. Suppose that Γ is (pointwise, uniformly) G-stable and
+(pointwise, uniformly) G-approximable. Then Γ is residually finite.
+21
+
+The hypothesis on G covers all cases above. When the groups in G are finite, this is clear,
+and when they are linear, this follows from Mal’cev’s Theorem [Mal40].
+Proof. We proceed with the proof without specifying the type of asymptotic homomorphisms,
+closeness, and approximability: the reader should read everything as pointwise, or everything
+as uniform.
+Let φ : Γ → G be an asymptotically injective asymptotic homomorphism. By stability,
+there exists a sequence of homomorphisms ψ : Γ → G which is asymptotically close to φ.
+Since φ is asymptotically injective, for each g ∈ Γ there exists N such that φn(g) ≥ ρ for all
+n ≥ N and some ρ = ρ(g) > 0. Up to taking a larger N, we also have that ψn(g) ≥ ρ/2, in
+particular ψn(g) ̸= 1. Since ψn(Γ) is a finitely generated subgroup of Gn ∈ G, it is residually
+finite by hypothesis, and so ψn(g) survives in some finite quotient of ψn(Γ). Since this is also
+a finite quotient of Γ, we conclude that Γ is residually finite.
+In the special case of pointwise stability and Thompson’s group F, we obtain the following
+more general version of a remark of Arzhantseva and Paunescu [AP15, Open problem]:
+Corollary 7.5. Let G be the family of symmetric groups with the normalized Hamming
+distance, the family of unitary groups with the Hilbert–Schmidt norm, or the family of unitary
+groups with the operatorn norm. If Thompson’s group F is pointwise G-stable, then it is not
+pointwise G-approximable, and in particular it is non-amenable.
+As we mentioned in the introduction, the amenability of Thompson’s group F is one of
+the most outstanding open problems in modern group theory.
+Proof. Thompson’s group F is not residually finite [CFP96]. So it follows from Proposition
+7.4 that it cannot be simultaneously pointwise G-stable and pointwise G-approximable. The
+last statement follows from the fact that amenable groups are sofic, hyperlinear, and MF.
+On the other hand, our results allow to settle the uniform approximability of Thompson’s
+groups, with respect to unitary groups and submultiplicative norms:
+Corollary 7.6. As usual, let U be the family of unitary groups equipped with submultiplicative
+norms. Then Thompson’s groups F, F ′, T and V are not uniformly U-approximable. The
+same holds for Γ ≀ Λ, whenever Λ is infinite and amenable, and Γ is non-abelian.
+We remark that Thompson’s groups T and V are generally regarded as good candidates
+for counterexamples to approximability problems.
+Proof. The statement for F, T and V follows from Theorem 1.2 and Proposition 7.4, together
+with the fact that they are not residually finite, and the statement for F ′ (which is not finitely
+generated) follows from the fact that F ′ contains a copy of F [CFP96]. The lamplighter case
+follows from Theorem 1.3 and Proposition 7.4, together with the fact that such lamplighters
+are not residually finite [Gru57].
+We do not know whether Thompson’s groups are uniformly G-approximable, when G is
+the family of unitary groups equipped with the Hilbert–Schmidt norm, and we conjecture
+that this is not the case. In the next section, we examine the case of symmetric groups via
+a more direct argument.
+22
+
+7.1
+Approximations by symmetric groups
+We end by proving, by a cohomology-free argument, that some of the groups studied in this
+paper are not uniformly approximable by symmetric groups, in a strong sense. For the rest
+of this section, we denote by S the family of symmetric groups equipped with the normalized
+Hamming distance. Our main result is an analogue of Corollary 5.2 for this approximating
+family (see Section 5.2 for the relevant definitions):
+Proposition 7.7. Let Γ be a proximal, boundedly supported group of orientation-preserving
+homeomorphisms of the line. Then every uniform asymptotic homomorphism φn : Γ′ →
+Skn ∈ S is uniformly asymptotically close to the trivial one. In particular, Γ′ is uniformly
+S-stable, and not uniformly S-approximable.
+The non-approximability follows from the fact that Γ′ is non-trivial (see Lemma 7.8).
+Note that for Γ as in the statement, Γ′ is simple [GG17, Theorem 1.1], so in particular every
+homomorphism Γ′ → Skn is trivial.
+The proof relies on known results on the flexible uniform stability of amenable groups
+[BC20] and uniform perfection of groups with proximal actions [GG17]. The finiteness of the
+groups in S will play a crucial role. We start with the following lemma:
+Lemma 7.8. Let Γ be as in Proposition 7.7. Then Γ′ is non-trivial, and the action of Γ′ on
+the line has no global fixpoints.
+Proof. If Γ′ is trivial, then Γ is abelian. This contradicts that the action is proximal and
+boundedly supported. Indeed, given g ∈ Γ, since g is centralized, the action of Γ on R must
+preserve the support of g, which is a proper subset of R. But then the action cannot be
+proximal.
+Now the set of global fixpoints of Γ′ is a closed subset X ⊂ R. Since Γ′ is normal in Γ,
+the action of Γ preserves X. But the action of Γ on R is proximal, in particular every orbit
+is dense, and since X is closed we obtain X = R. That is, Γ′ acts trivially on R. Since Γ
+is a subgroup of Homeo+(R), this implies that Γ′ is trivial, which contradicts the previous
+paragraph.
+We proceed with the proof:
+Proof of Proposition 7.7. It follows from [GG17, Theorem 1.1] that Γ′ is 2-uniformly perfect;
+that is, every element of Γ′ may be written as the product of at most 2 commutators (this
+uses the proximality hypothesis). Therefore it suffices to show that there exists a constant C
+such that for all g, h ∈ Γ′ it holds dkn(φn([g, h]), idkn) ≤ Cεn, where dkn denotes the Hamming
+distance on Skn and εn := def(φn). We drop the subscript n on φ and ε for clarity.
+Now let g, h ∈ Γ′, and let I, J ⊂ R be bounded intervals such that g is supported on I
+and h is supported on J. Since Γ′ acts without global fixpoints by Lemma 7.8, there exists
+t ∈ Γ′ such that t · inf(J) > sup(I). Since Γ′ is orientation-preserving, the same holds for
+all powers of t. In particular [g, tiht−i] = 1 for all i ≥ 1. Next, we apply [BC20, Theorem
+1.2] to the amenable group ⟨t⟩, to obtain an integer N such that kn ≤ N ≤ (1 + 1218ε)kn
+and a permutation τ in SN such that dN(φ(t)i, τ i) ≤ 2039ε for all i ∈ Z. Here dN denotes
+the normalized Hamming distance on the symmetric group SN, and φ is extended to a map
+23
+
+φ : Γ′ → SN with every φ(g) fixing each point in {kn + 1, . . . , N}.
+We compute (using
+τ N! = idN):
+dkn(φ([g, h]), idkn) ≤ dN(φ([g, h]), idN) ≤ dN([φ(g), φ(h)], idN) + O(ε)
+= dN([φ(g), τ N!φ(h)τ −N!], idN) + O(ε)
+≤ dN([φ(g), φ(tN!)φ(h)φ(t−N!)], idN) + O(ε)
+≤ dN(φ([g, tN!ht−N!]), idN) + O(ε)
+= dN(φ(1), idN) + O(ε) ≤ O(ε).
+Thus, there exists a constant C independent of g and h (C = 20000 suffices) such that
+dkn(φ([g, h]), idkn) ≤ Cε, which concludes the proof.
+Corollary 7.9. Consider the Thompson groups F ′, F, T.
+1. Every asymptotic homomorphism φn : F ′ → Skn ∈ S is uniformly asymptotically close
+to the trivial one.
+2. Every asymptotic homomorphism φn : F → Skn ∈ S is uniformly asymptotically close
+to one that factors through the abelianization.
+3. Every asymptotic homomorphism φn : T → Skn ∈ S is uniformly asymptotically close
+to the trivial one.
+Proof. Item 1. is an instance of Proposition 7.7: indeed F ′ satisfies the hypotheses for Γ,
+and F ′′ = F ′ since F ′ is simple. For Item 2., pick a section σ : Ab(F) → F, and define
+ψn(g) := φn(σ(Ab(g))). Using that ψn|F ′ is uniformly asymptotically close to the sequence
+of trivial maps, we obtain that φn and ψn are uniformly asymptotically close, and ψn factors
+as F → Ab(F)
+φn◦σ
+−−−→ Skn. Finally, Item 3. follows again from Item 1. and the fact that every
+element of T can be written as a product of two elements in isomorphic copies of F ′ (see the
+proof of Corollary 5.11).
+The corollary immediately implies that F, F ′ and T are not uniformly S-approximable,
+and that F ′ and T are uniformly S-stable. Since F has infinite abelianization, it follows from
+[BC20, Theorem 1.4] that it is not uniformly S-stable. However the corollary together with
+[BC20, Theorem 1.2] implies that it is flexibly uniformly S-stable; that is, every uniform
+asymptotic homomorphism is uniformly close to a sequence of homomorphisms taking values
+in a symmetric group of slightly larger degree. The case of Thompson’s group V can also be
+treated analogously (see the sketch of proof of Corollary 5.12).
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+27
+

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+arXiv:2301.08615v1  [hep-ph]  20 Jan 2023
+Photo-production of lowest Σ∗
+1/2− state within the Regge-effective Lagrangian approach
+Yun-He Lyu,1 Han Zhang,1 Neng-Chang Wei,2 Bai-Cian Ke,1 En Wang,1 and Ju-Jun Xie3, 2, 4
+1School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China
+2School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
+3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
+4Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics,
+Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China
+(Dated: January 23, 2023)
+Since the lowest Σ∗ state, with quantum numbers spin-parity JP = 1/2−, is far from estab-
+lished experimentally and theoretically, we have performed a theoretical study on the Σ∗
+1/2− photo-
+production within the Regge-effective Lagrangian approach. Taking into account that the Σ∗
+1/2−
+couples to the ¯KN channel, we have considered the contributions from the t-channel K exchange
+diagram. Moreover, these contributions from t-channel K∗ exchange, s-channel nucleon pole, u-
+channel Σ exchange, and the contact term, are considered. The differential and total cross sections
+of the process γn → K+Σ∗−
+1/2− are predicted with our model parameters. The results should be
+helpful to search for the Σ∗
+1/2− state experimentally in future.
+PACS numbers:
+I.
+INTRODUCTION
+The study of the low-lying excited Λ∗ and Σ∗ hyperon
+resonances is one of the most important issues in hadron
+physics.
+Especially, since the Λ(1405) was discovered
+experimentally [1, 2], its nature has called many atten-
+tions [3–8], and one explanation for Λ(1405) is that it
+is a ¯KN hadronic molecular state [9–14]. In addition,
+the isospin I = 1 partner of the Λ(1405), the lowest
+Σ∗
+1/2− is crucial to understand the light baryon spec-
+tra. At present, there is a Σ∗(1620) with JP = 1/2−
+listed in the latest version of Review of Particle Physics
+(RPP) [15]. It should be stressed that the Σ∗(1620) state
+is a one-star baryon resonance, and many studies indicate
+that the lowest Σ∗
+1/2− resonance is still far from estab-
+lished, and its mass was predicted to lie in the range of
+1380 ∼ 1500 MeV [13, 16–19]. Thus, searching for the
+lowest Σ∗
+1/2− is helpful to understand the low-lying ex-
+cited baryons with JP = 1/2− and the light flavor baryon
+spectra.
+The analyses of the relevant data of the process
+K−p → Λπ+π− suggest that there may exist a Σ∗
+1/2−
+resonance with mass about 1380 MeV [16, 17], which
+is consistent with the predictions of the unquenched
+quark models [20].
+The analyses of the K∗Σ photo-
+production also indicate that the Σ∗
+1/2− is possibly buried
+under the Σ∗(1385) peak with mass of 1380 MeV [21],
+and it is proposed to search for the Σ∗
+1/2− in the pro-
+cess Λc → ηπ+Λ [22]. A more delicate analysis of the
+CLAS data on the process γp → KΣπ [23] suggests that
+the Σ∗
+1/2− peak should be around 1430 MeV [13].
+In
+Refs. [24, 25], we suggest to search for such state in the
+processes of χc0(1P) → ¯ΣΣπ and χc0(1P) → ¯ΛΣπ. In
+addition, Ref. [26] has found one Σ∗
+1/2− state with mass
+around 1400 MeV by solving coupled channel scattering
+equations, and Ref. [27] suggests to search for this state
+in the photo-production process γp → K+Σ∗0
+1/2−.
+It’s worth mentioning that a Σ∗(1480) resonance with
+JP = 1/2− has been listed on the previous version of
+RPP [28].
+As early as 1970, the Σ∗(1480) resonance
+was reported in the Λπ+, Σπ, and p ¯K0 channels of the
+π+p scattering in the Princeton-Pennsylvania Accelera-
+tor 15-in.∼hydrogen bubble chamber [29, 30]. In 2004,
+a bump structure around 1480 MeV was observed in the
+K0
+Sp(¯p) invariant mass spectrum of the inclusive deep
+inelastic ep scattering by the ZEUS Collaboration [31].
+Furthermore, a signal for a resonance at 1480 ± 15 MeV
+with width of 60 ± 15 MeV was observed in the process
+pp → K+pY ∗0 [32]. Theoretically, the Σ∗(1480) was in-
+vestigated within different models [33–36]. In Ref. [36],
+the S-wave meson-baryon interactions with strangeness
+S = −1 were studied within the unitary chiral approach,
+and one narrow pole with pole position of 1468−i 13 MeV
+was found in the second Riemann sheet, which could be
+associated with the Σ∗(1480) resonance. However, the
+Σ∗(1480) signals are insignificant, and the existence of
+this state still needs to be confirmed within more precise
+experimental measurements.
+As we known, the photo-production reactions have
+been used to study the excited hyperon states Σ∗ and Λ∗,
+and the Crystal Ball [37–39], LEPS [40], and CLAS [23]
+Collaborations have accumulated lots of relevant exper-
+imental data.
+For instance, with these data, we have
+analyzed the process γp → KΛ∗(1405) to deepen the un-
+derstanding of the Λ∗(1405) nature in Ref. [41]. In order
+to confirm the existence of the Σ∗(1480), we propose to
+investigate the process γN → KΣ∗(1480) 1 within the
+1 Here after, we denote Σ∗(1480) as the lowest Σ∗
+1/2− state unless
+otherwise stated.
+
+2
+Regge-effective Lagrange approach.
+Considering the Σ∗(1480) signal was first observed
+in the π+Λ invariant mass distribution of the process
+π+p → π+K+Λ, and the significance is about 3 ∼
+4σ [30], we search for the charged Σ∗(1480) in the process
+γn → K+Σ∗−
+1/2−, which could also avoid the contribu-
+tions of possible excited Λ∗ states. We will consider the
+t-, s-, u-channels diagrams in the Born approximation
+by employing the effective Lagrangian approach, and the
+t-channel K/K∗ exchanges terms within Regge model.
+Then we will calculate the differential and total cross
+sections of the process γn → K+Σ∗−
+1/2− reaction, which
+are helpful to search for Σ∗
+1/2− experimentally.
+This paper is organized as follows. In Sec. II, the the-
+oretical formalism for studying the γn → K+Σ∗−(1480)
+reactions are presented. The numerical results of total
+and differential cross sections and discussion are shown
+in Sec. III. Finally, a brief summary is given in the last
+section.
+II.
+FORMALISM
+The reaction mechanisms of the Σ∗(1480) (≡ Σ∗)
+photo-production process are depicted in the Fig. 1,
+where we have taken into account the contributions from
+the t-channel K and K∗ exchange term, s-channel nu-
+cleon pole term, u-channel Σ exchange term, and the
+contact term, respectively.
+γ(k1)
+K(k2)
+N(p1)
+Σ∗(p2)
+K, K∗
+γ
+K
+N
+Σ∗
+N
+γ
+Σ∗
+N
+K
+γ
+K
+N
+Σ∗
+Σ
+(a)
+(b)
+(c)
+(d)
+FIG. 1: The mechanisms of the γn → K+Σ∗−
+1/2− process. (a)
+t-channel K/K∗ exchange terms, (b) s-channel nuclear term,
+(c) u-channel Σ exchange term, and (d) contact term. The
+k1, k2, p1, and p2 stand for the four-momenta of the initial
+photon, kaon, neutron, and Σ∗(1480), respectively.
+To compute the scattering amplitudes of the Feynman
+diagrams shown in Fig. 1 within the effective Lagrangian
+approach, we use the Lagrangian densities for the elec-
+tromagnetic and strong interaction vertices as used in
+Refs. [27, 42–46]
+LγKK = −ie
+�
+K† (∂µK) −
+�
+∂µK†�
+K
+�
+Aµ,
+(1)
+LγKK∗ = gγKK∗ǫµναβ∂µAν∂αK∗
+βK,
+(2)
+LγNN = −e ¯N
+�
+γµˆe −
+ˆκN
+2MN
+σµν∂ν
+�
+AµN,
+(3)
+LγΣΣ∗ = eµΣΣ∗
+2MN
+¯Σγ5σµν∂νAµΣ∗ + h.c.,
+(4)
+LKNΣ = −igKNΣ ¯Nγ5ΣK + h.c.,
+(5)
+LK∗NΣ∗ = igK∗NΣ∗
+√
+3
+¯K∗µ ¯Σ∗γµγ5N + h.c.
+(6)
+LKNΣ∗ = gKNΣ∗ ¯K ¯
+Σ∗N + h.c.,
+(7)
+where e(=
+√
+4πα) is the elementary charge unit, Aµ is the
+photon filed, and ˆe ≡ (1+τ3)/2 denotes the charge opera-
+tor acting on the nucleon field. ˆκN ≡ κpˆe+κn(1−ˆe) is the
+anomalous magnetic moment, and we take κn = −1.913
+for neutron [15]. MN and MΣ denote the masses of nu-
+cleon and the ground-state of Σ hyperon, respectively.
+The strong coupling gKNΣ is taken to be 4.09 from
+Ref. [47].
+The gγKK∗ = 0.254 GeV−1 is determined
+from the experimental data of ΓK∗→K+γ [15] and the
+value of gK∗NΣ∗ = −3.26 − i0.06 is taken from Ref [26].
+In addition, the coupling gKNΣ∗ = 8.74 GeV is taken
+from Ref. [36], and the transition magnetic moment
+µΣΣ∗ = 1.28 is taken from Ref. [27]
+With the effective interaction Lagrangian densities
+given above, the invariant scattering amplitudes are de-
+fined as
+M = ¯uΣ∗(p2, sΣ∗)Mµ
+huN(k2, sp)ǫµ(k1, λ),
+(8)
+where uΣ∗ and uN stand for the Dirac spinors, respec-
+tively, while ǫµ(k1, λ) is the photon polarization vector
+and the sub-indice h corresponds to different diagrams
+of Fig. 1. The reduced amplitudes Mµ
+h are written as
+Mµ
+K∗ =
+egγKK∗gK∗NΣ∗
+√
+3MK∗(t − M 2
+K∗)ǫαβµνk1αk2βγνγ5,
+(9)
+Mµ
+K− = −2iegKNΣ∗
+t − M 2
+K
+kµ
+2 ,
+(10)
+Mµ
+Σ− = −i
+eµΣΣ∗gKNΣ
+2Mn(u − M 2
+Σ∗)(q/u − MΣ)σµνk1ν, (11)
+Mµ
+n =
+κngKNΣ∗
+2Mn(s − M 2n)σµνk1ν(q/s + Mn).
+(12)
+In order to keep the full photoproduction amplitudes
+considered here gauge invariant, we adopt the amplitude
+of the contact term
+Mµ
+c = −iegKNΣ∗
+pµ
+2
+p2 · k1
+,
+(13)
+for γn → K+Σ∗−
+1/2−.
+
+3
+It is known that the Reggeon exchange mechanism
+plays a crucial role at high energies and forward an-
+gles [48–51], thus we will adopt Regge model for mod-
+eling the t-channel K and K∗ contributions by replacing
+the usual pole-like Feynman propagator with the corre-
+sponding Regge propagators as follows,
+1
+t − M 2
+K
+→ FRegge
+K
+=
+� s
+sK
+0
+�αK(t)
+πα′
+K
+sin(παK(t))Γ(1 + αK(t)),(14)
+1
+t − M 2
+K∗
+→ FRegge
+K∗
+=
+� s
+sK∗
+0
+�αK∗(t)
+πα′
+K∗
+sin(παK∗(t))Γ(αK∗(t)),(15)
+with αK(t) = 0.7 GeV−2 × (t − M 2
+K) and αK∗(t) = 1 +
+0.83 Gev−2 × (t − M 2
+K∗) the linear Reggeon trajectory.
+The constants sK
+0 and sK∗
+0
+are determined to be 3.0 GeV2
+and 1.5 GeV2, respectively [52]. Here, the α′
+K and α′
+K∗
+are the Regge-slopes.
+Then, the full photo-production amplitudes for γn →
+K+Σ∗−
+1/2− reaction can be expressed as
+Mµ =
+�
+Mµ
+K− + Mµ
+c
+� �
+t − M 2
+K−
+�
+FRegge
+K
++ Mµ
+Σ−fu
++ Mµ
+K∗
+�
+t − M 2
+K∗
+�
+FRegge
+K∗
++ Mµ
+nfs,
+(16)
+While FRegge
+K
+and FRegge
+K∗
+stand for the Regge propaga-
+tors. The form factors fs and fu are included to suppress
+the large momentum transfer of the intermediate par-
+ticles and describe their off-shell behavior, because the
+intermediate hadrons are not point-like particles.
+For
+s-channel and u-channel baryon exchanges, we use the
+following form factors [42, 53]
+fi(q2
+i ) =
+�
+Λ4
+i
+Λ4
+i + (q2
+i − M 2
+i )2
+�2
+, i = s, u
+(17)
+with Mi and qi being the masses and four-momenta of
+the intermediate baryons, and the Λi is the cut-off values
+for baryon exchange diagrams.
+In this work, we take
+Λs = Λu = 1.5 GeV, and will discuss the results with
+different cut-off.
+Finally, the unpolarized differential cross section in the
+center of mass (c.m.) frame for the γn → KΣ∗−
+1/2− reac-
+tion reads
+dσ
+dΩ = MNMΣ∗|⃗kc.m.
+1
+||⃗pc.m.
+1
+|
+8π2(s − M 2
+N)2
+�
+λ,sp,sΣ∗
+|M|2,
+(18)
+where s denotes the invariant mass square of the center
+of mass (c.m.) frame for Σ∗
+1/2− photo-production. Here
+⃗kc.m.
+1
+and ⃗pc.m.
+1
+are the three-momenta of the photon and
+K meson in the c.m.
+frame, while dΩ = 2πdcosθc.m.,
+with θc.m. the polar outgoing K scattering angle.
+III.
+NUMERICAL RESULTS AND
+DISCUSSIONS
+In this section, we show our numerical results of the dif-
+ferential and total cross sections for the γn → K+Σ∗−
+1/2−
+reaction.
+The masses of the mesons and baryons are
+taken from RPP [15], as given in Table I. In addition, the
+mass and width of the Σ(1480) are M = 1480 ± 15 GeV
+and Γ = 60 ± 15 GeV, respectively [28].
+TABLE I: Particle masses used in this work.
+Particle
+Mass (MeV)
+n
+939.565
+Σ−
+1197.449
+K+
+493.677
+K−
+493.677
+K∗
+891.66
+First we show the angle dependence of the differential
+cross sections for the γn → K+Σ∗−
+1/2− reaction in Fig. 2,
+where the the center-of-mass energies W = √s varies
+from 2.0 to 2.8 GeV. The black curves labeled as ‘Total’
+show the results of all the contributions from the t-, s-,
+u-channels, and contact term. The blue-dot curves and
+red-dashed curves stand for the contributions from the
+u-channel Σ exchange and t-channel K exchange mecha-
+nism, respectively. The magenta-dot-dashed curves and
+the green-dot curves correspond to the contributions
+from the s-channel and t-channel K∗ exchange diagrams,
+respectively, while the cyan-dot-dashed curves represent
+the contribution from the contact term. According to the
+differential cross sections, one can find that the t-channel
+K meson exchange term plays an important role at for-
+ward angles for the process γn → K+Σ∗−
+1/2−, mainly due
+to the Regge effects of the t-change K exchange. The
+K-Reggeon exchange shows steadily increasing behavior
+with cosθc.m. and falls off drastically at very forward an-
+gles. In addition, the u-channel Σ exchange term mainly
+contribute to the backward angles for both processes.
+It should be stressed that the contribution from the t-
+channel K∗ exchange term is very small and could be
+safely neglected for the process γn → K+Σ∗−
+1/2−, which
+is consistent with the results of Ref. [27].
+In addition to the the differential cross sections, we
+have also calculated the total cross section of the γn →
+K+Σ∗−
+1/2− reaction as a function of the initial photon en-
+ergy. The results are shown in Fig. 3. The black curve
+labeled as ‘Total’ shows the results of all the contribu-
+tions, including t-, s-, u- channels and contact term. The
+blue-dot and red-dashed curves stand for the contribu-
+tions from the u- channel Σ exchange and t- channel
+K exchange mechanism, respectively. The magenta-dot-
+dashed and the green-dot curves show the contribution of
+s-channel and t-channel K∗ exchange diagrams, respec-
+tively, while the cyan-dot-dashed curve represents the
+
+4
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+dσ/dcosθc.m. (µb)
+cosθc.m.
+W=2.0 GeV
+K-t
+K*-t
+s-channel
+u-channel
+contact term
+Total
+W=2.1 GeV
+W=2.2 GeV
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+W=2.3 GeV
+W=2.4 GeV
+W=2.5 GeV
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+-1
+-0.5
+0
+0.5
+1
+W=2.6 GeV
+-1
+-0.5
+0
+0.5
+1
+W=2.7 GeV
+-1
+-0.5
+0
+0.5
+1
+W=2.8 GeV
+FIG. 2: (Color online) γn → K+Σ∗−
+1/2− differential cross sections as a function of cosθc.m. are plotted for γn-invariant mass
+intervals (in GeV units). The black curve labeled as ‘Total’ shows the results of all the contributions, including t-, s-, u- channels
+and contact term. The blue-dot and red-dashed curves stand for the contributions from the effective Lagrangian approach u-
+channel Σ exchange and t- channel K exchange mechanism, respectively. The magenta-dot-dashed and the green-dot-dashed
+curves show the contribution of s-channel and t-channel K∗ exchange diagrams, respectively, while the cyan-dot-dashed curve
+represent the contribution of the contact term.
+contribution of the contact term. For the γn → K+Σ∗−
+1/2−
+reaction its total cross section attains a maximum value
+of about 4.3 µb at Eγ = 2.3 GeV. It is expected that the
+Σ∗(1480) could be observed by future experiments in the
+process γn → K+Σ∗− (1480) → Σ−π0/Σ0π−/Σ−γ.
+Finally, we also show the total cross section for γn →
+K+Σ∗−
+1/2− with the cut-off Λs/u = 1.2, 1.5, and 1.8 GeV
+in Fig. 4, where one can find the total cross sections are
+weakly dependence on the value of the cut-off. Since the
+precise couplings of the Σ(1480) are still unknown, the
+
+5
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+ 5.5
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+σ (µb)
+Eγ (GeV)
+K-t
+K*-t
+s-channel
+u-channel
+contact term
+Total
+FIG. 3:
+(Color online) Total cross section for γn
+→
+K+Σ∗
+1/2− is plotted as a function of the lab energy Eγ. The
+black curve labeled as ‘Total’ shows the results of all the con-
+tributions, including t-,s-,u- channels and contact term. The
+blue-dot and red-dashed curves stand for the contributions
+from the effective Lagrangian approach u- channel Σ exchange
+and t- channel K exchange mechanism, respectively.
+The
+magenta-dot-dashed and the green-dot curves show the con-
+tribution of s-channel and t-channel K∗ exchange diagrams,
+respectively, while the cyan-dot-dashed curve represents the
+contribution of the contact term.
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+ 5.5
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+σ (µb)
+Eγ (GeV)
+Λs,u = 1.2 GeV
+Λs,u = 1.5 GeV
+Λs,u = 1.8 GeV
+FIG. 4:
+(Color online) Total cross section for γn
+→
+K+Σ∗
+1/2− with the cut-off Λs/u = 1.2, 1.5, and 1.8 GeV.
+future experiment would be helpful to constrain these
+couplings if the state Σ(1480) is confirmed.
+IV.
+SUMMARY
+The lowest Σ∗−
+1/2− is far from established, and its ex-
+istence is important to understand the low-lying excited
+baryon with JP = 1/2−. There are many experimen-
+tal hints of the Σ∗(1480), which has been listed in the
+previous version of the Review of Particle Physics. We
+propose to search for this state in the photoproduction
+process to confirm its existence.
+Assuming that the JP
+=
+1/2− low lying state
+Σ∗ (1480) has a sizeable coupling to the ¯KN according
+the study of Ref. [36], we have phenomenologically inves-
+tigated the γn → K+Σ∗−
+1/2− reaction by considering the
+contributions from the t-channel K/K∗ exchange term,
+s-channel nucleon term, u-channel Σ exchange term, and
+contact term within the Regge-effective Lagrange ap-
+proach.
+The differential cross sections and total cross
+sections for these processes are calculated with our model
+parameters. The total cross section of γn → K+Σ∗−
+1/2−
+is about 4.3 µb around Eγ = 2.3 GeV. We encourage
+our experimental colleagues to measure γn → K+Σ∗−
+1/2−
+process.
+Acknowledgements
+This
+work
+is
+supported
+by
+the
+National
+Natu-
+ral Science Foundation of China under Grant Nos.
+12192263, 12075288, 11735003, and 11961141012, the
+Natural Science Foundation of Henan under Grand No.
+222300420554.
+It is also supported by the Project of
+Youth Backbone Teachers of Colleges and Universities
+of Henan Province (2020GGJS017), the Youth Talent
+Support Project of Henan (2021HYTP002), the Open
+Project of Guangxi Key Laboratory of Nuclear Physics
+and Nuclear Technology, No.NLK2021-08, the Youth In-
+novation Promotion Association CAS.
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='03520v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='FA]  9 Jan 2023  CLASSIFYING WEAK PHASE RETRIEVAL  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' AKRAMI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We will give several surprising equivalences and consequences of  weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' These results give a complete understanding of the dif-  ference between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We also answer two  longstanding open problems on weak phase retrieval: (1) We show that the  families of weak phase retrievable frames {xi}m  i=1 in Rn are not dense in the  family of m-element sets of vectors in Rn for all m ≥ 2n − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' (2) We show  that any frame {xi}2n−2  i=1  containing one or more canonical basis vectors in Rn  cannot do weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We provide numerous examples to show that  the obtained results are best possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Introduction  The concept of frames in a separable Hilbert space was originally introduced by  Duffin and Schaeffer in the context of non-harmonic Fourier series [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Frames  are a more flexible tool than bases because of the redundancy property that make  them more applicable than bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Phase retrieval is an old problem of recovering  a signal from the absolute value of linear measurement coefficients called intensity  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Phase retrieval and norm retrieval have become very active areas of  research in applied mathematics, computer science, engineering, and more today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Phase retrieval has been defined for both vectors and subspaces (projections) in all  separable Hilbert spaces, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', [3], [4], [5], [6], [9], [10] and [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  The concept of weak phase retrieval weakened the notion of phase retrieval and it  has been first defined for vectors in ([8] and [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The rest of the paper is organized  as follows: In Section 2, we give the basic definitions and certain preliminary results  to be used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Weak phase retrieval by vectors is introduced in section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' In section 4 we show that any family of vectors {xi}2n−2  i=1  doing weak phase  retrieval cannot contain a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' In section 5, we show that the weak phase  retrievable frames are not dense in all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' And in section 6 we give several  surprising equivalences and consequences of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' These results  give a complete understanding of the difference between weak phase retrieval and  phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' preliminaries  First we give the background material needed for the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let H be a finite  or infinite dimensional real Hilbert space and B(H) the class of all bounded linear  operators defined on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The natural numbers and real numbers are denoted by  “N” and “R”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We use [m] instead of the set {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', m} and use  [{xi}i∈I] instead of span{xi}i∈I, where I is a finite or countable subset of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' 42C15, 42C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Real Hilbert frames, Full spark, Phase retrieval, Weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  The first author was supported by NSF DMS 1609760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  1  2  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' AKRAMI  denote by Rn a n dimensional real Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We start with the definition of a  real Hilbert space frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A family of vectors {xi}i∈I in a finite or infinite dimensional separable  real Hilbert space H is a frame if there are constants 0 < A ≤ B < ∞ so that  A∥x∥2 ≤  �  i∈I  |⟨x, xi⟩|2 ≤ B∥x∥2,  for all  f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  The constants A and B are called the lower and upper frame bounds for {xi}i∈I,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If an upper frame bound exists, then {xi}i∈I is called a B-Bessel  seqiemce or simply Bessel when the constant is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If A = B, it is called an  A-tight frame and in case A = B = 1, it is called a Parseval frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The values  {⟨x, xi⟩}∞  i=1 are called the frame coefficients of the vector x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  It is immediate that a frame must span the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We will need to work with  Riesz sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A family X = {xi}i∈I in a finite or infinite dimensional real Hilbert  space H is a Riesz sequence if there are constants 0 < A ≤ B < ∞ satisfying  A  �  i∈I  |ci|2 ≤ ∥  �  i∈I  cixi∥2 ≤ B  �  i∈I  |ci|2  for all sequences of scalars {ci}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If it is complete in H, we call X a Riesz basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  For an introduction to frame theory we recommend [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Throughout the paper the orthogonal projection or simply projection will be a self-  adjoint positive projection and {ei}∞  i=1 will be used to denote the canonical basis  for the real space Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', a basis for which  ⟨ei, ej⟩ = δi,j =  �  1  if i = j,  0  if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A family of vectors {xi}i∈I in a real Hilbert space H does phase  (norm) retrieval if whenever x, y ∈ H, satisfy  |⟨x, xi⟩| = |⟨y, xi⟩|  for all i ∈ I,  then x = ±y  (∥x∥ = ∥y∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Phase retrieval was introduced in reference [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' See reference [1] for an introduc-  tion to norm retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Note that if {xi}i∈I does phase (norm) retrieval, then so does {aixi}i∈I for any  0 < ai < ∞ for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' But in the case where |I| = ∞, we have to be careful to  maintain frame bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' This always works if 0 < infi∈I ai ≤ supi∈Iai < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' But  this is not necessary in general [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The complement property is an essential issue  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A family of vectors {xi}i∈I in a finite or infinite dimensional real  Hilbert space H has the complement property if for any subset J ⊂ I,  either span{xi}i∈J = H  or  span{xi}i∈Jc = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Fundamental to this area is the following for which the finite dimensional case  appeared in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  WEAK PHASE RETRIEVAL  3  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A family of vectors {xi}i∈I does phase retrieval in Rn if and only if it  has the complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  We recall:  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A family of vectors {xi}m  i=1 in Rn is full spark if for every I ⊂  [m] with |I| = n , {xi}i∈I is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If {xi}m  i=1 does phase retrieval in Rn, then m ≥ 2n− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If m = 2n− 1,  {xi}m  i=1 does phase retrieval if and only if it is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  We rely heavily on a significant result from [2]:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If {xi}2n−2  i=1  does weak phase retrieval in Rn then for every I ⊂ [2n−2],  if x ⊥ span{xi}i∈I and y ⊥ {xi}i∈Ic then  x  ∥x∥ +  y  ∥y∥ and  x  ∥x∥ −  y  ∥y∥ are disjointly  supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' In particular, if ∥x∥ = ∥y∥ = 1, then x + y and x − y are disjointly  supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Hence, if x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , an) then y = (ǫ1a1, ǫ2a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , ǫnan), where  ǫi = ±1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The above theorem may fail if ∥x∥ ̸= ∥y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' For example, consider the  weak phase retrievable frame in R3:  \uf8ee  \uf8ef\uf8ef\uf8f0  1  1  1  −1  1  1  1  −1  1  1  1  −1  \uf8f9  \uf8fa\uf8fa\uf8fb  Also, x = (0, 1, −1) is perpendicular to rows 1 and 2 and y = (0, 1  2, 1  2) is orthogonal  to rows 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' But x + y = (0, 3  2, 1  2) and x − y = (0, −1  2 , −3  2 ) and these are not  disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' But if we let them have the same norm we get x = (0, 1, −1)  and y = (0, 1, 1) so x + y = (0, 1, 0) and x − y = (0, 0, 1) and these are disjointly  supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Weak phase retrieval  The notion of “Weak phase retrieval by vectors” in Rn was introduced in [8] and  was developed further in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' One limitation of current methods used for retrieving  the phase of a signal is computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Recall that a generic family of (2n − 1)-  vectors in Rn satisfies phaseless reconstruction, however no set of (2n − 2)-vectors  can (See [7] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' By generic we are referring to an open dense set in the set  of (2n − 1)-element frames in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Two vectors x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , bn) in Rn weakly  have the same phase if there is a |θ| = 1 so that phase(ai) = θphase(bi) for all  i ∈ [n], for which ai ̸= 0 ̸= bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  If θ = 1, we say x and y weakly have the same signs and if θ = −1, they weakly  have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Therefore with above definition the zero vector in Rn weakly has the same phase  with all vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' For x ∈ R, sgn(x) = 1 if x > 0 and sgn(x) = −1 if x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A family of vectors {xi}m  i=1 does weak phase retrieval in Rn if for  any x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , bn) in Rn with |⟨x, xi⟩| = |⟨y, xi⟩| for  all i ∈ [m], then x and y weakly have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  A fundamental result here is  4  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' AKRAMI  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' [8] Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , bn) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  The  following are equivalent:  (1) We have sgn(aiaj) = sgn(bibj), for all 1 ≤ i ̸= j ≤ n  (2) Either x, y have weakly the same sign or they have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  It is clear that if {xi}m  i=1 does weak phase retrieval in Rn, then {cixi}m  i=1 does  weak phase retrieval as long as ci > 0 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  The following appears in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If X = {xi}m  i=1 does weak phase retrieval in Rn, then m ≥ 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Finally, we have:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' [7] If a frame X = {xi}2n−2  i=1  does weak phase retrieval in Rn, then X  is a full spark frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Clearly the converse of above theorem is not hold, for example {(1, 0), (0, 1)} is  full spark frame that fails weak phase retrieval in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  If {xi}i∈I does phase retrieval and R is an invertible operator on the space  then {Rxi}i∈I does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' This follows easily since |⟨x, Rxi⟩| = |⟨y, Rxi⟩|  implies |⟨R∗x, xi⟩| = |⟨R∗y, xi⟩|, and so R∗x = θR∗y for |θ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Since R is  invertible, x = θy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' This result fails badly for weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' For example,  let e1 = (1, 0), e2 = (0, 1), x1 = ( 1  √  2,  1  √  2, x2 = ( 1  √  2, −1  √  2) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then {e1, e2} fails  weak phase retrieval, {x1, x2} does weak phase retrieval and Uei = xi is a unitary  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Frames Containing Unit Vectors  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Any frame {xi}2n−2  i=1  whith one or more canonical basis vectors in Rn  cannot do weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We proceed by way of contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Recall that {xi}2n−2  i=1  must be full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Let {ei}n  i=1 be the canonical orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Assume I ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', 2n−2}  with |I| = n − 1 and assume x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , bn) with ∥x∥ =  ∥y∥ = 1 and x ⊥ X = span{xi}i∈I and y ⊥ span{xi}2n−2  i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' After reindexing {ei}n  i=1  and {xi}2n−2  i=1 }, we assume x1 = e1, I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', n−1 and Ic = {n, n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , 2n−  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Since ⟨x, x1⟩ = a1 = 0, by Theorem 2, b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let P be the projection on  span{ei}n  i=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So {Pxi}2n−2  i=n  is (n − 1)-vectors in an (n − 1)-dimensional space and  y is orthogonal to all these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So there exist {ci}2n−2  i=n  not all zero so that  2n−2  �  i=n  ciPxi = 0 and so  2n−1  �  i=n  cixi(1)x1 −  2n−2  �  i=n  cixi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  That is, our vectors are not full spark, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The fact that there are (2n− 2) vectors in the theorem is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' For  example, e1, e2, e1 + e2 is full spark in R2, so it does phase retrieval - and hence  weak phase retrieval - despite the fact that it contains both basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  The converse of Theorem 5 is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Consider the full spark frame X = {(1, 2, 3), (0, 1, 0), (0, −2, 3), (1, −2, −3)}  in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Every set of its two same coordinates,  {(1, 2), (0, 1), (0, −2), (1, −2)}, {(1, 3), (0, 0), (0, 3), (1, −3)}, and  WEAK PHASE RETRIEVAL  5  {(2, 3), (1, 0), (−2, 3), (−2, −3)}  do weak phase retrieval in R2, but by Theorem 5, X cannot do weak phase retrieval  in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Weak Phase Retrievable Frames are not Dense in all Frames  If m ≥ 2n − 1 and {xi}m  i=1 is full spark then it has complement property and  hence does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Since the full spark frames are dense in all frames, it  follows that the frames doing phase retrieval are dense in all frames with ≥ 2n − 1  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We will now show that this result fails for weak phase retrievable frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  The easiest way to get very general frames failing weak phase retrieval is:  Choose x, y ∈ Rn so that x + y, x − y do not have the same or opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Let X1 = x⊥ and Y1 = y⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then span{X1, X2} = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Choose {xi}n−1  i=1 vectors  spanning X1 and {xi}2n−2  i=n  be vectors spanning X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then {xi}2n−2  i=1  is a frame for  Rn with x ⊥ xi, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', n − 1 and y ⊥ xi, for all i = n, n + 1, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  It follows that  |⟨x + y, xi⟩| = |⟨x − y, xi⟩|, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , n,  but x, y do not have the same or opposite signs and so {xi}2n−2  i=1  fails weak phase  retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If X is a subspace of Rn, we define the sphere of X as  SX = {x ∈ X : ∥x∥ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If X, Y are subspaces of Rn, we define the distance between X  and Y as  d(X, Y ) = supx∈SXinfy∈SY ∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  It follows that if d(X, Y ) < ǫ then for any x ∈ X there is a z ∈ SY so that  ∥ x  ∥x∥ − z∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Letting y = ∥x∥z we have that ∥y∥ = ∥x∥ and ∥x − y∥ < ǫ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let X, Y be hyperplanes in Rn and unit vectors x ⊥ X, y ⊥ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If  d(X, Y ) < ǫ then min{∥x − y∥, ∥x + y∥} < 6ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Since span{y, Y } = Rn, x = ay + z for some z ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' By replacing y by −y  if necessary, we may assume 0 < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' By assumption, there is some w ∈ X with  ∥w∥ = ∥z∥ so that ∥w − z∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Now  a = a∥y∥ = ∥ay∥ = ∥x − z∥ ≥ ∥x − w∥ − ∥w − z∥ ≥ ∥x∥ − ǫ = 1 − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  So, 1 − a < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Also, 1 = ∥x∥2 = a2 + ∥w∥2 implies a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' 0 < 1 − a < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  1 = ∥x∥2 = ∥ay + z∥2 = a2∥y∥2 + ∥z∥2 = a2 + ∥z∥2 ≥ (1 − ǫ)2 + ∥z∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  So  ∥z∥2 ≤ 1 − (1 − ǫ)2 = 2ǫ − ǫ2 ≤ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Finally,  ∥x − y∥2 = ∥(ay + z) − y∥2  ≤ (∥(1 − a)y∥ + ∥z∥)2  ≤ (1 − a)2∥y∥2 + ∥z∥2 + 2(1 − a)∥y∥∥z∥  < ǫ2 + 2ǫ + 2  √  2ǫ2  < 6ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  6  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' AKRAMI  □  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let X, Y be hyperplanes in Rn, {xi}n−1  i=1 be a unit norm basis for X and  {yi}n−1  i=1 be a unit norm basis for Y with basis bounds B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If �n−1  i=1 ∥xi − yi∥ < ǫ  then d(X, Y ) < 2ǫB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let 0 < A ≤ B < ∞ be upper and lower basis bounds for the two bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Given a unit vector x = �n−1  i=1 aixi ∈ X, let y = �n−1  i=1 aiyi ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We have that  sup1≤i≤n−1|ai| ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We compute:  ∥x − y∥ = ∥  n−1  �  i=1  ai(xi − yi)∥  ≤  n−1  �  i=1  |ai|∥xi − yi∥  ≤ (sup1≤i≤n−1|ai|)  n−1  �  i=1  ∥xi − yi∥ ≤ Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  So  ∥y∥ ≥ ∥x∥ − ∥x − y∥ ≥ 1 − Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  ����x −  y  ∥y∥  ���� ≤ ∥x − y∥ +  ����y −  y  ∥y∥  ����  ≤ Bǫ +  1  ∥y∥∥(1 − ∥y∥)y∥  = Bǫ + (1 − ∥y∥)  ≤ 2Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  It follows that d(X, Y ) < 2Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let {xi}n  i=1 be a basis for Rn with unconditional basis constant B and  assume yi ∈ Rn satisfies �n  i=1 ∥xi − yi∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then {yi}n  i=1 is a basis for Rn which  is 1 + ǫB-equivalent to {xi}n  i=1 and has unconditional basis constant B(1 + ǫB)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Fix {ai}n  i=1 and compute  ∥  n  �  i=1  aiyi∥ ≤ ∥  n  �  i=1  aixi∥ + ∥  n  �  i=1  |ai|(xi − yi)∥  ≤ ∥  n  �  i=1  aixi∥ + (sup1≤i≤n|ai|)  n  �  i=1  ∥xi − yi∥  ≤ ∥  n  �  i=1  aixi∥ + (sup1≤i≤n|ai|)ǫ  ≤ ∥  n  �  i=1  aixi∥ + ǫB∥  n  �  i=1  aixi∥  = (1 + ǫB)∥  n  �  i=1  aixi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  WEAK PHASE RETRIEVAL  7  Similarly,  ∥  n  �  i=1  |ai|yi∥ ≥ (1 − ǫB)∥  n  �  i=1  aixi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  So {xi}n  i=1 is (1 + ǫB)-equivalent to {yi}n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  For ǫi = ±1,  ∥  n  �  i=1  ǫiaiyi∥ ≤ (1 + ǫB)∥  n  �  i=1  ǫiaixi∥  ≤ B(1 + ǫB)∥  n  �  i=1  aixi∥  ≤ B(1 + ǫB)2∥  n  �  i=1  aiyi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  and so {yi}n  i=1 is a B(1 + ǫB) unconditional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The family of m-element weak phase retrieval frames are not dense in  the set of m-element frames in Rn for all m ≥ 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We may assume m = 2n−2 since for larger m we just repeat the (2n-2) vec-  tors over and over until we get m vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let {ei}n  i=1 be the canonical orthonormal  basis for Rn and let xi = ei for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' By [10], there is an orthonormal  sequence {xi}2n−2  i=n+1 so that {xi}2n−2  i=1  is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let  X = span{xi}n−1  i=1 and Y = span{xi}2n−2  i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Then x = en ⊥ X and there is a  ∥y∥ = 1 with y ⊥ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Note that ⟨x − y, en⟩ ̸= 0 ̸= ⟨x + y, en⟩, for otherwise,  x = ±y ⊥ span{xi}i̸=n, contradicting the fact that the vectors are full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So  there is a j = n and a δ > 0 so that |(x + y)(j)|, |(x − y)(j)| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We will show  that there exists an 0 < ǫ so that whenever {yi}2n−2  i=1  are vectors in Rn satisfying  �n  i=1 ∥xi − yi∥ < ǫ, then {yi}n  i=1 fails weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Fix 0 < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Assume {yi}2n−2  i=1  are vectors so that �2n−2  i=1  ∥xi−yi∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Choose unit  vectors x′ ⊥ span{yi}i∈I, y′ ⊥ span{yi}i∈Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' By Proposition 2 and Lemma 1, we  may choose ǫ small enough (and change signs if necessary) so that ∥x−x′∥, ∥y−y′∥ <  δ  4B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Hence, since the unconditional basis constant is B,  |[(x + y) − (x′ + y′)](j)|  ≤ |(x − x′)j| + |(y − y′)(j)|  < B∥x − x′∥ + B∥y − y′∥  ≤ 2B δ  4B = δ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  It follows that  |(x′ + y′)(j)| ≥ |(x + y)(j)| − |[(x + y) − (x′ + y′)](j)| ≥ δ − 1  2δ = δ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Similarly, |(x′ − y′)(j)| > δ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So x′ + y′, x′ − y′ are not disjointly supported and so  {yi}2n−2  i=1  fails weak phase retrieval by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Classifying Weak Phase Retrieval  In this section we will give several surprising equivalences and consequences of  weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' These results give a complete understanding of the difference  between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Now we give a surprising and very strong classification of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  8  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' AKRAMI  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let {xi}2n−2  i=1  be non-zero vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The following are equivalent:  (1) The family {xi}2n−2  i=1  does weak phase retrieval in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (2) If x, y ∈ Rn and  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='1)  |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , 2n − 2,  then one of the following holds:  (a) x = ±y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (b) x and y are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' (1) ⇒ (2): Given the assumption in the theorem, assume (a) fails and we will  show that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Since {xi}2n−2  i=1  does weak phase retrieval, replacing y by −y if necessary, Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='1 implies  aj = bj whenever aj ̸= 0 ̸= bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Let  I = {1 ≤ i ≤ 2n − 2 : ⟨x, xi⟩ = ⟨y, yi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Then  x + y ⊥ xi for all i ∈ Ic and x − y ⊥ xi for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  By Theorem 2,  x + y  ∥x + y +  x − y  ∥x − y∥ and  x + y  ∥x + y∥ −  x − y  ∥x − y∥ are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Assume there is a 1 ≤ j ≤ n with aj = bj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then  (x + y)(j)  ∥x + y∥  + (x − y)(j)  ∥x − y∥  =  2aj  ∥x + y∥ and (x + y)(j)  ∥x + y∥  − (x − y)(j)  ∥x − y∥  =  2aj  ∥x + y∥,  Contradicting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (2) ⇒ (1): This is immediate since (a) and (b) give the conditions for weak phase  retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  Phase retrieval is when (a) in the theorem holds for every x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So this the-  orem shows clearly the difference between weak phase retrieval and phase retrieval:  namely when (b) holds at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If {xi}2n−2  i=1  does weak phase retrieval in Rn, then there are disjointly  supported non-zero vectors x, y ∈ Rn satisfying:  |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Since {xi}2n−2  i=1  must fail phase retrieval, (b) of Theorem 7 must hold at least  once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let {ei}n  i=1 be the canonical orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If J ⊂ [n],  we define PJ as the projection onto span{ei}i∈J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let {xi}m  i=1 be unit vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The following are equivalent:  (1) Whenever I ⊂ [2n − 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic,  there is no j ∈ [n] so that ⟨x, ej⟩ = 0 = ⟨y, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (2) For every J ⊂ [n] with |J| = n − 1, {Pjxi}2n−2  i=1  does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (3) For every J ⊂ [n] with |J| < n, {PJxi}2n−2  i=1  does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  WEAK PHASE RETRIEVAL  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' (1) ⇒ (2): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So assume (2) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then choose  J ⊂ [n] with |J| = n − 1, J = [n] \\ {j}, and {PJxi}2n−2  i=1  fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' In  particular, it fails complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' That is, there exists I ⊂ [2n− 2] and span  {PJxi}i∈I ̸= PJRn and span {Pjxi}i∈Ic ̸= PJRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So there exists norm one vectors  x, y in PJRn with PJx = x ⊥ PJxi for all i ∈ I and PJy = y ⊥ PJxi for all i ∈ Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Extend x, y to all of Rn by setting x(j) = y(j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Hence, x ⊥ xi for i ∈ I and  y ⊥ xi for i ∈ Ic, proving (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (2) ⇒ (3): This follows from the fact that every projection of a set of vectors  doing phase retrieval onto a subset of the basis also does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (3) ⇒ (2): This is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  (3) ⇒ (1): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So assume (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then there is a  I ⊂ [2n− 2] and 0 ̸= x ⊥ xi for i ∈ I and 0 ̸= y ⊥ xi for i ∈ Ic and a j ∈ [n] so that  ⟨x, ej⟩ = ⟨y, ej⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' It follows that x = PJx, y = PJy are non zero and x ⊥ Pjxi  for all i ∈ I and y ⊥ Pjxi for i ∈ Ic, so {PJxi}2n−2  i=1  fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' The assumptions in the theorem are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' That is, in general,  {xi}m  i=1 can do weak phase retrieval and {PJxi}m  i=1 may fail phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' For  example, in R3 consider the row vectors {xi}4  i=1 of:  \uf8ee  \uf8ef\uf8ef\uf8f0  1  1  1  −1  1  1  1  −1  1  1  1  −1  \uf8f9  \uf8fa\uf8fa\uf8fb  This set does weak phase retrieval, but if J = {2, 3} then x = (0, 1, −1) ⊥ PJxi for  i = 1, 2 and y = (0, 1, 1) ⊥ xi for i = 3, 4 and {PJxi}4  i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Assume {xi}2n−2  i=1  does weak phase retrieval in Rn and for every J ⊂ [n]  {PJxi}2n−2  i=1  does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Then if x, y ∈ Rn and  |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' , 2n − 2,  then there is a J ⊂ [n] so that  x(j) =  �  aj ̸= 0 for j ∈ J  0 for j ∈ Jc  y(j) =  �  0 for j ∈ J  bj ̸= 0 for j ∈ Jc  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Let {ei}n  i=1 be the unit vector basis of Rn and for I ⊂ [n], let PI be  the projection onto XI = span{ei}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' For every m ≥ 1, there are vectors {xi}m  i=1  so that for every I ⊂ [1, n], {PIxi}m  i=1 is full spark in XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We do this by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' For m=1, let x1 = (1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' This  satisfies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' So assume the theorem holds for {xi}m  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Choose I ⊂ [1, n]  with |I| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Choose J ⊂ I with |J| = k − 1 and let XJ = span{xi}i∈J ∪ {xi}i∈Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Then XJ is a hyperplane in Rn for every J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Since there only exist finitely many  such J′s there is a vector xm+1 /∈ XJ for every J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' We will show that {xi}m+1  i=1  satisfies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Let I ⊂ [1, n] and J ⊂ I with |J| = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' If PIxm+1 /∈ XJ, then {PIxi}i∈J is  linearly independent by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' On the other hand, if m + 1 ∈ J  then xm+1 /∈ XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' But, if PIxm+1 ∈ span{PIxi}i∈J\\m+1, since (I − PI)xm+1 ∈  span{ei}i∈Ic, it follows that xm+1 ∈ XJ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  □  10  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' AKRAMI  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' In the above proposition, none of the xi can have a zero coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Since if it does, projecting the vectors onto that coordinate produces a zero vector  and so is not full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  References  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Akrami, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Casazza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Hasankhani Fard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Rahimi, A note on norm retrievable  real Hilbert space frames, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' (517)2, (2023) 126620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Casazza, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Akrami, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content=' Rahimi, fundamental results on weak phase retrieval, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
+page_content='  Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+Higher order Bernstein-B´ezier and N´ed´elec finite elements for the
+relaxed micromorphic model
+Adam Sky1,
+Ingo Muench2,
+Gianluca Rizzi3
+and
+Patrizio Neff4
+January 5, 2023
+Abstract
+The relaxed micromorphic model is a generalized continuum model that is well-posed in the space
+X = [H 1]3 × [H (curl)]3. Consequently, finite element formulations of the model rely on H 1-conforming
+subspaces and N´ed´elec elements for discrete solutions of the corresponding variational problem. This work
+applies the recently introduced polytopal template methodology for the construction of N´ed´elec elements.
+This is done in conjunction with Bernstein-B´ezier polynomials and dual numbers in order to compute hp-
+FEM solutions of the model. Bernstein-B´ezier polynomials allow for optimal complexity in the assembly
+procedure due to their natural factorization into univariate Bernstein base functions. In this work, this
+characteristic is further augmented by the use of dual numbers in order to compute their values and their
+derivatives simultaneously. The application of the polytopal template methodology for the construction of
+the N´ed´elec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein-
+B´ezier basis. We introduce the Bernstein-B´ezier basis along with its factorization to univariate Bernstein
+base functions, the principle of automatic differentiation via dual numbers and a detailed construction of
+N´ed´elec elements based on Bernstein-B´ezier polynomials with the polytopal template methodology. This is
+complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on
+the consistent coupling condition. The performance of the elements is shown in examples of the relaxed
+micromorphic model.
+Key words: N´ed´elec elements, Bernstein-B´ezier elements, relaxed micromorphic model, dual numbers, au-
+tomatic differentiation, hp-FEM, generalized continua.
+1
+Introduction
+One challenge that arises in the computation of materials with a pronounced micro-structure is the necessity of
+modelling the complex geometry of the domain as a whole, in order to correctly capture its intricate kinematics.
+In other words, unit-cell geometries in metamaterials or various hole-shapes in porous media have to be accounted
+for in order to assert the viability of the model. Naturally, this correlates with the resolution of the discretization
+in finite element simulations, resulting in longer computation times.
+The relaxed micromorphic model [35] offers an alternative approach by introducing a continuum model with
+enriched kinematics, accounting for the independent distortion arising from the micro-structure. As such, for
+each material point, the model introduces the microdistortion field P in addition to the standard displacement
+field u. Consequently, each material point is endowed with twelve degrees of freedom, effectively turning into
+an affine-deformable micro-body with its own orientation. In contrast to the classical micromorphic model [17]
+1Corresponding author: Adam Sky, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund,
+August-Schmidt-Str. 8, 44227 Dortmund, Germany, email: adam.sky@tu-dortmund.de
+2Ingo Muench, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt-Str.
+8, 44227 Dortmund, Germany, email: ingo.muench@tu-dortmund.de
+3Gianluca Rizzi, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt-
+Str. 8, 44227 Dortmund, Germany, email: gianluca.rizzi@tu-dortmund.de
+4Patrizio Neff,
+Chair for Nonlinear Analysis and Modelling, Faculty of Mathematics, Universit¨at Duisburg-Essen, Thea-
+Leymann Str. 9, 45127 Essen, Germany, email: patrizio.neff@uni-due.de
+1
+arXiv:2301.01491v1  [math.NA]  4 Jan 2023
+
+by Eringen [15] and Mindlin [29], the relaxed micromorphic model does not employ the full gradient of the
+microdistortion DP in its energy functional but rather its skew-symmetric part Curl P , designated as the
+micro-dislocation. Therefore, the micro-dislocation Curl P remains a second-order tensor, whereas DP is a
+third-order tensor.
+Further, the model allows the transition between materials with a pronounced micro-
+structure and homogeneous materials using the characteristic length scale parameter Lc, which governs the
+influence of the micro-structure. In highly homogeneous materials the characteristic length scale parameter
+approaches zero Lc → 0, and for materials with a pronounced micro-structure its value is related to the size of
+the underlying unit-cell geometry. Recent works demonstrate the effectiveness of the model in the simulation
+of band-gap metamaterials [7, 10, 13, 27, 28] and shielding against elastic waves [4, 40, 41, 46].
+Furthermore,
+analytical solutions are already available for bending [43], torsion [42], shear [44], and extension [45] kinematics.
+We note that the usage of the curl operator in the free energy functional directly influences the appropriate
+Hilbert spaces for existence and uniqueness of the related variational problem. Namely, the relaxed micromorphic
+model is well-posed in {u, P } ∈ X = [H 1]3 × [H (curl)]3 [18,34], although the regularity of the microdistortion
+can be improved to P ∈ [H 1]3×3 for certain smoothness of the data [22, 38]. As shown in [52], the X -space
+asserts well-posedness according to the Lax-Milgram theorem, such that H 1-conforming subspaces and N´ed´elec
+elements [9,30,31] inherit the well-posedness property as well.
+In this work we apply the polytopal template methodology introduced in [50] in order to construct higher
+order N´ed´elec elements based on Bernstein polynomials [23] and apply the formulation to the relaxed micro-
+morphic model. Bernstein polynomials are chosen due to their optimal complexity property in the assembly
+procedure [1]. We further enhance this feature by employing dual numbers [16] in order to compute the values of
+the base functions and their derivatives simultaneously. The polytopal template methodology allows to extend
+this property to the assembly of the N´ed´elec base functions, resulting in fast computations. Alternatively, the
+formulation of higher order elements on the basis of Legendre polynomials can be found in [48, 54, 58]. The
+construction of low order N´ed´elec elements can be found in [5, 51] and specifically in the context of the the
+relaxed micromorphic model in [47,49,52,53].
+This paper is structured as follows. First, we introduce the relaxed micromorphic model and its limit cases
+with respect to the characteristic length scale parameter Lc, after which we reduce it to a model of antiplane
+shear [55]. Next, we shortly discuss Bernstein polynomials and dual numbers for automatic differentiation. The
+B´ezier polynomial basis for triangles and tetrahedra is introduced, along with its factorization, highlighting
+its compatibility with dual numbers. We consider a numerical example in antiplane shear for two-dimensional
+elements, a three-dimensional example for convergence of cylindrical bending, and a benchmark for the behaviour
+of the model with respect to the characteristic length scale parameter Lc. Lastly, we present our conclusions
+and outlook.
+The following definitions are employed throughout this work:
+• vectors are indicated by bold letters. Non-bold letters represent scalars;
+• in general, formulas are defined using the Cartesian basis, where the base vectors are denoted by e1, e2
+and e3;
+• three-dimensional domains in the physical space are denoted with V ⊂ R3. The corresponding reference
+domain is given by Ω;
+• analogously, in two dimensions we employ A ⊂ R2 for the physical domain and Γ for the reference domain;
+• curves on the physical domain are denoted by s, whereas curves in the reference domain by µ;
+• the tangent and normal vectors in the physical domain are given by t and n, respectively. Their counter-
+parts in the reference domain are τ for tangent vectors and ν for normal vectors.
+2
+
+2
+The relaxed micromorphic model
+The relaxed micromorphic model [35] is governed by a free energy functional, incorporating the gradient of the
+displacement field Du, the microdistortion P and the Curl of the microdistortion
+I(u, P ) = 1
+2
+�
+V
+⟨sym(Du − P ), Ce sym(Du − P )⟩ + ⟨sym P , Cmicro sym P ⟩
++ ⟨skew(Du − P ), Cc skew(Du − P )⟩ + µmacroL2
+c⟨Curl P , L Curl P ⟩ dV
+−
+�
+V
+⟨u, f⟩ + ⟨P , M⟩ dV → min
+w.r.t.
+{u, P } ,
+(2.1)
+where the Curl operator for second order tensors is defined row-wise as
+Curl P =
+�
+�
+curl(
+�P11
+P12
+P13
+�
+)
+curl(
+�P21
+P22
+P23
+�
+)
+curl(
+�
+P31
+P32
+P33
+�
+)
+�
+� =
+�
+�
+P13,y − P12,z
+P11,z − P13,x
+P12,x − P11,y
+P23,y − P22,z
+P21,z − P23,x
+P22,x − P21,y
+P33,y − P32,z
+P31,z − P33,x
+P32,x − P31,y
+�
+� ,
+curl p = ∇ × p ,
+p : V ⊂ R3 → R3 ,
+(2.2)
+and curl(·) is the vectorial curl operator. The displacement field and the microdistortion field are functions of
+the reference domain
+u : V ⊂ R3 → R3 ,
+P : V ⊂ R3 → R3×3 .
+(2.3)
+The tensors Ce, Cmicro, L ∈ R3×3×3×3 are standard positive definite fourth order elasticity tensors. For isotropic
+materials they take the form
+Ce = λe1 ⊗ 1 + 2µe J ,
+Cmicro = λmicro1 ⊗ 1 + 2µmicro J .
+(2.4)
+where 1 is the second order identity tensor and J is the fourth order identity tensor. The fourth order tensor
+Cc ∈ R3×3×3×3 is a positive semi-definite material tensor related to Cosserat micro-polar continua and accounts
+for infinitesimal rotations Cc : so(3) → so(3), where so(3) is the space of skew-symmetric matrices.
+For isotropic materials there holds Cc = 2µc J, where µc ≥ 0 is called the Cosserat couple modulus. Further,
+for simplicity, we assume L = J in the following. The macroscopic shear modulus is denoted by µmacro and
+Lc represents the characteristic length scale motivated by the geometry of the microstructure. The forces and
+micro-moments are given by f and M, respectively.
+Equilibrium is found at minima of the energy functional, which is strictly convex (also for Cc ≡ 0). As such,
+we consider variations with respect to its parameters, namely the displacement and the microdistortion. Taking
+variations of the energy functional with respect to the displacement field u yields
+δuI =
+�
+V
+⟨sym Dδu, Ce sym(Du − P )⟩ + ⟨skew Dδu, Cc skew(Du − P )⟩ − ⟨δu, f⟩ dV = 0 .
+(2.5)
+The variation with respect to the microdistortion P results in
+δP I =
+�
+V
+⟨sym δP , Ce sym(Du − P )⟩ + ⟨skew δP , Cc skew(Du − P )⟩
+− ⟨sym δP , Cmicro sym P ⟩ − µmacroL2
+c⟨Curl δP , Curl P ⟩ + ⟨δP , M⟩ dV = 0 .
+(2.6)
+From the total variation we extract the bilinear form
+a({δu, δP }, {u, P }) =
+�
+V
+⟨sym(Dδu − δP ), Ce sym(Du − P )⟩ + ⟨sym δP , Cmicro sym P ⟩
++ ⟨skew(Dδu − δP ), Cc skew(Du − P )⟩ + µmacroL2
+c⟨Curl δP , Curl P ⟩ dV ,
+(2.7)
+and linear form of the loads
+l({δu, δP }) =
+�
+V
+⟨δu, f⟩ + ⟨δP , M⟩ dV .
+(2.8)
+3
+
+Applying integration by parts to Eq. (2.5) yields
+�
+∂V
+⟨δu , [Ce sym(Du − P ) + Cc skew(Du − P )] n⟩ dA
+−
+�
+V
+⟨δu , Div[Ce sym(Du − P ) + Cc skew(Du − P )] − f⟩ dV = 0 .
+(2.9)
+Likewise, integration by parts of Eq. (2.6) results in
+�
+V
+⟨δP , Ce sym(Du − P ) + Cc skew(Du − P ) − Cmicro sym P − µmacroL2
+c Curl Curl P + M⟩ dV
+− µmacroL2
+c
+�
+∂V
+⟨δP , Curl P × n⟩ dA = 0 .
+(2.10)
+The strong form is extracted from Eq. (2.9) and Eq. (2.10) by splitting the boundary
+A = AD ∪ AN ,
+AD ∩ AN = ∅ ,
+(2.11)
+into a Dirichlet boundary with embedded boundary conditions and a Neumann boundary with natural boundary
+conditions, such that no tractions are imposed on the Neumann boundary
+− Div[Ce sym(Du − P ) + Cc skew(Du − P )] = f
+in
+V ,
+(2.12a)
+−Ce sym(Du − P ) − Cc skew(Du − P ) + Cmicro sym P + µmacro L2
+c Curl Curl P = M
+in
+V ,
+(2.12b)
+u = �u
+on
+Au
+D ,
+(2.12c)
+P × n = �P × n
+on
+AP
+D , (2.12d)
+[Ce sym(Du − P ) + Cc skew(Du − P )] n = 0
+on
+Au
+N ,
+(2.12e)
+Curl P × n = 0
+on
+AP
+N .
+(2.12f)
+The force stress tensor �σ := Ce sym(Du − P ) + Cc skew(Du − P ) is symmetric if and only if Cc ≡ 0, a case
+which is permitted. Problem. 2.12 represents a tensorial Maxwell-problem coupled to linear elasticity. We
+observe that the Dirichlet boundary condition for the microdistortion controls only its tangential components.
+It is unclear, how to control the micro-movements of a material point without also affecting the displacement.
+Therefore, the relaxed micromorphic model introduces the so called consistent coupling condition [11]
+P × n = D�u × n
+on
+AP
+D ,
+(2.13)
+where the prescribed displacement on the Dirichlet boundary �u automatically dictates the tangential component
+of the microdistortion on that same boundary. Consequently, the consistent coupling condition enforces the
+definitions AD = Au
+D = AP
+D and AN = Au
+N = AP
+N (see Fig. 2.1). Further, the consistent coupling condition
+substitutes Eq. (2.12d).
+The set of equations in Problem. 2.12 remains well-posed for Cc ≡ 0 due to the
+generalized Korn inequality for incompatible tensor fields [24–26,36]. The inequality relies on a non-vanishing
+Dirichlet boundary for the microdistortion field AP
+D ̸= ∅, which the consistent coupling condition guarantees.
+2.1
+Limits of the characteristic length scale parameter - a true two scale model
+In the relaxed micromorphic model the characteristic length Lc takes the role of a scaling parameter between
+the well-defined macro and the micro scales. This property, unique to the relaxed micromorphic model, allows
+the theory to interpolate between materials with a pronounced micro-structure and homogeneous materials,
+thus relating the characteristic length scale parameter Lc to the size of the micro-structure in metamaterials.
+In the lower limit Lc → 0 the continuum is treated as homogeneous and the solution of the classical Cauchy
+continuum theory is retrieved [3,32]. This can be observed by reconsidering Eq. (2.12b) for Lc = 0,
+−Ce sym(Du − P ) − Cc skew(Du − P ) + Cmicro sym P = M ,
+(2.14)
+which can now be used to express the microdistortion P algebraically
+sym P = (Ce + Cmicro)−1(sym M + Ce sym Du) ,
+skew P = C−1
+c
+skew M + skew Du .
+(2.15)
+4
+
+x
+y
+n
+f
+M
+V
+AD = Au
+D = AP
+D
+AN = Au
+N = AP
+N
+Figure 2.1: The domain in the relaxed micromorphic model with Dirichlet and Neumann boundaries under
+internal forces and micro-moments. The Dirichlet boundary of the microdistortion is given by the consistent
+coupling condition. The model can capture the complex kinematics of an underlying micro-structure.
+Setting M = 0 corresponds to Cauchy continua, where micro-moments are not accounted for. Thus, one finds
+Cc skew(Du − P ) = 0 ,
+Ce sym(Du − P ) = Cmicro sym P ,
+sym P = (Ce + Cmicro)−1Ce sym Du .
+(2.16)
+Applying the former results to Eq. (2.12a) yields
+− Div[Ce sym(Du − P )] = − Div[Cmicro(Ce + Cmicro)−1Ce sym Du] = − Div[Cmacro sym Du] = f ,
+(2.17)
+where the definition
+Cmacro = Cmicro(Ce + Cmicro)−1Ce
+(2.18)
+relates the meso- and micro-elasticity tensors to the classical macro-elasticity tensor of the Cauchy continuum.
+In fact, Cmacro contains the material constants that arise from standard homogenization for large periodic
+structures [3,32]. For isotropic materials one can directly express the macro parameters [33]
+µmacro =
+µe µmicro
+µe + µmicro
+,
+2µmacro + 3λmacro =
+(2µe + 3λe)(2µmicro + 3λmicro)
+(2µe + 3λe) + (2µmicro + 3λmicro)
+(2.19)
+in terms of the parameters of the relaxed micromorphic model.
+In the upper limit Lc → +∞, the stiffness of the micro-body becomes dominant. As the characteristic
+length Lc can be viewed as a zoom-factor into the microstructure, the state Lc → +∞ can be interpreted as the
+entire domain being the micro-body itself. However, this is only theoretically possible as in practice, the limit is
+given by the size of one unit cell. Since the energy functional being minimized contains µmacroL2
+c∥ Curl P ∥2, on
+contractible domains and bounded energy this implies the reduction of the microdistortion to a gradient field
+P → Dv due to the classical identity
+Curl Dv = 0
+∀ v ∈ [C ∞(V )]3 ,
+(2.20)
+thus asserting finite energies of the relaxed micromorphic model for arbitrarily large characteristic length values
+Lc. The corresponding energy functional in terms of the reduced kinematics {u, v} : V → R3 now reads
+I(u, v) = 1
+2
+�
+V
+⟨sym(Du − Dv), Ce sym(Du − Dv)⟩ + ⟨sym Dv, Cmicro sym Dv⟩
++ ⟨skew(Du − Dv), Cc skew(Du − Dv)⟩ dV −
+�
+V
+⟨u, f⟩ + ⟨Dv, M⟩ dV ,
+(2.21)
+such that variation with respect to the two vector fields u and v leads to
+δuI =
+�
+V
+⟨sym Dδu, Ce sym(Du − Dv)⟩ + ⟨skew Dδu, Cc skew(Du − Dv)⟩ − ⟨δu, f⟩ dV = 0 ,
+(2.22a)
+δvI =
+�
+V
+⟨sym Dδv, Ce sym(Du − Dv)⟩ + ⟨skew Dδv, Cc skew(Du − Dv)⟩
+− ⟨sym Dδv, Cmicro sym Dv⟩ + ⟨Dδv, M⟩ dV = 0 .
+(2.22b)
+5
+
+The resulting bilinear form is given by
+a({δu, δv}, {u, v}) =
+�
+V
+⟨sym(Dδu − Dδv), Ce sym(Du − Dv)⟩ + ⟨sym Dδv, Cmicro sym Dv⟩
++ ⟨skew(Dδu − Dδv), Cc skew(Du − Dv)⟩ dV .
+(2.23)
+By partial integration of Eq. (2.22a) and Eq. (2.22b) one finds the equilibrium equations
+− Div[Ce sym(Du − Dv) + Cc skew(Du − Dv)] = f
+in
+V ,
+(2.24a)
+− Div[Ce sym(Du − Dv) + Cc skew(Du − Dv)] + Div[Cmicro sym Dv] = Div M
+in
+V .
+(2.24b)
+We can now substitute the right-hand side of Eq. (2.24a) into Eq. (2.24b) to find
+− Div(Cmicro sym Dv) = f − Div M .
+(2.25)
+Clearly, setting v = u satisfies both local equilibrium equations Eq. (2.24a) and Eq. (2.24b) for f = 0. Further,
+the consistent coupling condition Eq. (2.13) is also automatically satisfied, asserting the equivalence of the
+tangential projections of both fields on the boundary of the domain.
+Since, as shown in [32, 52] using the
+extended Brezzi theorem, the case Lc → +∞ is well-posed (including Cc ≡ 0), the solution v = u is the unique
+solution to the bilinear form Eq. (2.23) with the right-hand side
+l({δu, δv}) = ⟨Dδv, M⟩ dV .
+(2.26)
+Effectively, equation Eq. (2.25) implies that the limit Lc → +∞ defines a classical Cauchy continuum with a
+finite stiffness governed by Cmicro, representing the upper limit of the stiffness for the relaxed micromorphic
+continuum [32], where the corresponding forces read m = Div M. We emphasize that this interpretation of
+Cmicro is impossible in the classical micromorphic model since there the limit Lc → +∞ results in a constant
+microdistortion field P : V → R3×3 as its full gradient DP is incorporated via µmacroL2
+c∥DP ∥2 into the energy
+functional [6].
+2.2
+Antiplane shear
+We introduce the relaxed micromorphic model of antiplane shear1 [55] by reducing the displacement field to
+u =
+�0,
+0,
+u�T ,
+(2.27)
+such that u = u(x, y) is a function of the x − y-plane. Consequently, its gradient reads
+Du =
+�
+�
+0
+0
+0
+0
+0
+0
+u,x
+u,y
+0
+�
+� .
+(2.28)
+The structure of the microdistortion tensor is chosen accordingly
+P =
+�
+�
+0
+0
+0
+0
+0
+0
+p1
+p2
+0
+�
+� ,
+Curl P =
+�
+�
+0
+0
+0
+0
+0
+0
+0
+0
+p2,x − p1,y
+�
+� =
+�
+�
+0
+0
+0
+0
+0
+0
+0
+0
+curl2Dp
+�
+� .
+(2.29)
+1Note that the antiplane shear model encompasses 1 + 2 = 3 degrees of freedom and is the simplest non-trivial active version
+for the relaxed micromorphic model, as the one-dimensional elongation ansatz features only 1 + 1 = 2 degrees of freedom and
+eliminates the curl operator
+I(u, p) = 1
+2
+�
+s
+(λe + 2µe)|u′ − p|2 + (λmicro + 2µmicro)|p|2 ds −
+�
+s
+u f + p m ds → min
+w.r.t.
+{u, p} ,
+since Du = u′ e1 ⊗ e1 and P = p e1 ⊗ e1, such that skew(Du − P ) = 0 and Curl P = 0. This is not to be confused with uniaxial
+extension, which entails 1 + 3 = 4 degrees of freedom [45].
+6
+
+Analogously to the displacement field u, the microdistortion P is also set to be a function of the {x, y}-variables
+P = P (x, y). We observe the following sym-skew decompositions of the gradient and microdistortion tensors
+sym P = 1
+2
+�
+�
+0
+0
+p1
+0
+0
+p2
+p1
+p2
+0
+�
+� ,
+sym(Du − P ) = 1
+2
+�
+�
+0
+0
+u,x − p1
+0
+0
+u,y − p2
+u,x − p1
+u,y − p2
+0
+�
+� ,
+skew(Du − P ) = 1
+2
+�
+�
+0
+0
+p1 − u,x
+0
+0
+p2 − u,y
+u,x − p1
+u,y − p2
+0
+�
+� .
+(2.30)
+Clearly, there holds
+tr[sym P ] = tr[sym(Du − P )] = tr[skew(Du − P )] = 0 ,
+(2.31)
+such that the contraction with the material tensors reduces to
+Ce sym(Du − P ) = 2µe sym(Du − P ) ,
+Cmicro sym(Du − P ) = 2µmicro sym P ,
+Cc skew(Du − P ) = 2µc skew(Du − P ) .
+(2.32)
+As such, the quadratic forms of the energy functional are given by
+⟨sym(Du − P ), Ce sym(Du − P )⟩ = µe∥∇u − p∥2 ,
+(2.33a)
+⟨skew(Du − P ), Cc skew(Du − P )⟩ = µc∥∇u − p∥2 ,
+(2.33b)
+⟨sym P , Cmicro sym P ⟩ = µmicro∥p∥2 ,
+(2.33c)
+with the definitions
+∇u =
+�u,x
+u,y
+�
+,
+p =
+�p1
+p2
+�
+.
+(2.34)
+The resulting energy functional for antiplane shear reads therefore
+I(u, p) = 1
+2
+�
+A
+(µe + µc)∥∇u − p∥2 + µmicro∥p∥2 + µmacroL2
+c∥curl2Dp∥2 dA −
+�
+A
+u f + ⟨p, m⟩ dA .
+(2.35)
+In order to maintain consistency with the three-dimensional model we must choose µc = 0. The reasoning for
+this choice is explained upon in Remark 2.1 (see also Fig. 2.2). Consequently, the energy functional is given by
+I(u, p) = 1
+2
+�
+A
+µe∥∇u − p∥2 + µmicro∥p∥2 + µmacroL2
+c∥curl2Dp∥2 dA
+−
+�
+A
+u f + ⟨p, m⟩ dA → min
+w.r.t.
+{u, p} .
+(2.36)
+Note that on two-dimensional domains the differential operators are reduced to
+∇u =
+�u,x
+u,y
+�
+,
+R∇u =
+� u,y
+−u,x
+�
+,
+R =
+�
+0
+1
+−1
+0
+�
+,
+curl2Dp = div(Rp) = p2,x − p1,y ,
+(2.37)
+where we note that curl2D is just a rotated divergence. Taking variations of the energy functional with respect
+to the displacement field results in
+δuI =
+�
+A
+µe⟨∇δu, ∇u − p⟩ − δu f dA = 0 ,
+(2.38)
+and variation with respect to the microdistortion yields
+δpI =
+�
+A
+µe⟨δp, ∇u − p⟩ − µmicro⟨δp, p⟩ − µmacroL2
+c(curl2Dδp)curl2Dp + ⟨δp, m⟩ dA = 0 .
+(2.39)
+7
+
+Consequently, one finds the bilinear and linear forms
+a({δu, δp}, {u, p}) =
+�
+A
+µe⟨∇δu − δp, ∇u − p⟩ + µmicro⟨δp, p⟩ + µmacroL2
+c(curl2Dδp)curl2Dp dA ,
+(2.40a)
+l({δu, δp}) =
+�
+A
+δu f + ⟨δp, m⟩ dA .
+(2.40b)
+Partial integration of Eq. (2.38) results in
+�
+∂A
+δu ⟨µe(∇u − p), n⟩ ds −
+�
+A
+δu [µe div(∇u − p) + f] dA = 0 ,
+(2.41)
+and analogously for Eq. (2.39), yielding
+�
+A
+⟨δp, µe(∇u − p) − µmicro p − µmacroL2
+cR∇curl2Dp + m⟩ dA −
+�
+∂A
+⟨δp, µmacroL2
+c(curl2Dp) t⟩ ds = 0 . (2.42)
+Consequently, the strong form reads
+−µe div(∇u − p) = f
+in
+A ,
+(2.43a)
+−µe(∇u − p) + µmicro p + µmacroL2
+cR∇curl2Dp = m
+in
+A ,
+(2.43b)
+u = �u
+on
+su
+D ,
+(2.43c)
+⟨p, t⟩ = ⟨�p, t⟩
+on
+sP
+D ,
+(2.43d)
+⟨∇u, n⟩ = ⟨p, n⟩
+on
+su
+N ,
+(2.43e)
+curl2Dp = 0
+on
+sP
+N .
+(2.43f)
+The consistent coupling condition accordingly reduces to
+⟨p, t⟩ = ⟨∇�u, t⟩
+on
+sD = sP
+D = su
+D .
+(2.44)
+Remark 2.1
+Note that without setting µc = 0 in the antiplane shear model, the analogous result to Eq. (2.17) in the limit
+Lc → 0 would read
+−
+� µmicro [µe + µc]
+µe + µc + µmicro
+�
+�
+��
+�
+̸=µmacro
+∆u = f ,
+(2.45)
+where the relation to the macro parameter µmacro in Eq. (2.19) is lost. Further, the limit defined in Eq. (2.16)
+with M = 0 yields the contradiction
+sym P = (Ce + Cmicro)−1Ce sym Du ,
+Cc skew P = Cc skew Du ,
+(2.46)
+since the equations degenerate to
+p =
+µe
+µe + µmicro
+∇u ,
+µcp = µc∇u ,
+(2.47)
+due to the equivalent three-dimensional forms for antiplane shear. Choosing µmicro = 0 leads to a loss of structure
+in the strong form Problem. 2.43, while satisfying Eq. (2.47). As such, we must set the Cosserat couple modulus
+µc = 0 to preserve the structure of the equations and satisfy both Eq. (2.19) and Eq. (2.47).
+Although the relaxed micromorphic model includes the Cosserat model as a singular limit for Cmicro → +∞
+(µmicro → +∞), it is impossible to deduce the Cosserat model of antiplane shear as a limit of the antiplane
+relaxed micromorphic model, since one needs to satisfy Eq. (2.47) for µc > 0 and µmicro → +∞, which is
+impossible.
+8
+
+The kinematic reduction of the relaxed micromorphic model to antiplane shear and its behaviour in the limit
+cases of its material parameters is depicted in Fig. 2.2.
+relaxed micromorphic
+Cosserat elasticity
+linear elasticity
+with Cmacro
+antiplane relaxed
+micromorphic
+antiplane Cosserat
+elasticity
+antiplane linear
+elasticity
+with µmacro
+Lc → 0
+Cmicro → +∞ ,
+µc > 0
+Lc → 0 ,
+µc ��� 0
+µmicro → +∞ ,
+µc > 0
+(contradiction)
+antiplane
+shear
+antiplane
+shear
+antiplane
+shear
+antiplane linear
+elasticity
+with µmicro
+linear elasticity
+with Cmicro
+Lc → +∞
+Lc → +∞
+two-scale
+model
+two-scale
+model
+non-
+commutative
+Figure 2.2: Kinematic reduction of the relaxed micromorphic model to antiplane shear and consistency at limit
+cases according to Remark 2.1 and Section 2.1. The two-scale nature of the relaxed micromorphic model can
+be clearly observed.
+3
+Polynomial basis
+In this section we briefly introduce Bernstein polynomials and dual numbers. Bernstein polynomials are used
+to construct both the H 1-conforming subspace and, in conjunction with the polytopal template methodology,
+the N´ed´elec elements. The computation of derivatives of the Bernstein base functions is achieved by employing
+dual numbers, thus enabling the calculation of the value and the derivative of a base function simultaneously.
+3.1
+Bernstein polynomials
+Bernstein polynomials of order p are given by the binomial expansion of the barycentric representation of the
+unit line
+1 = (λ1 + λ2)p = ((1 − ξ) + ξ)p =
+p
+�
+i=0
+�p
+i
+�
+ξi(1 − ξ)p−i =
+p
+�
+i=0
+p!
+i!(p − i)!ξi(1 − ξ)p−i ,
+(3.1)
+9
+
+b4
+0(ξ)
+b4
+1(ξ)
+b4
+2(ξ)
+b4
+3(ξ)
+b4
+4(ξ)
+ξ
+1
+1
+0
+1
+1/2
+1/2
+Figure 3.1: Bernstein base functions of degree p = 4 on the unit domain. Their sum forms a partition of unity.
+The base functions are symmetric for ξ = 0.5 with respect to their indices and always positive.
+where ξ ∈ [0, 1]. The Bernstein polynomial reads
+bp
+i (ξ) =
+�p
+i
+�
+ξi(1 − ξ)p−i .
+(3.2)
+A direct result of the binomial expansion is that Bernstein polynomials form a partition of unity, see also Fig. 3.1
+p
+�
+i=0
+bp
+i (ξ) = 1 .
+(3.3)
+Another consequence is that Bernstein polynomials are non-negative and less than or equal to 1
+0 ≤ bp
+i (ξ) ≤ 1 ,
+ξ ∈ [0, 1] .
+(3.4)
+A necessary condition for the use of Bernstein polynomials in finite element approximations is for them to span
+the entire polynomial space.
+Theorem 3.1 (Span of Bernstein polynomials)
+The span of Bernstein polynomials forms a basis of the one-dimensional polynomial space
+Pp(ξ) = span{bp
+i } ,
+ξ ⊆ R .
+(3.5)
+Proof. First we observe
+dim(span{bp
+i }) = dim Pp(ξ) = p + 1 .
+(3.6)
+The proof of linear independence is achieved by contradiction. Let the set span{bp
+i } with 0 < i ≤ p be linearly
+dependent, then there exists some combination with at least one non-zero constant ci ̸= 0 such that
+p
+�
+i=1
+cibp
+i (ξ) = 0 ,
+d
+dξ
+p
+�
+i=1
+cibp
+i (ξ) = 0 .
+(3.7)
+However, by the partition of unity property Eq. (3.3), only the full combination (0 ≤ i ≤ p) generates a constant
+and by the exact sequence property the kernel of the differentiation operator is exactly the space of constants
+ker(∂) = R. The linear independence of the full span also follows from the partition of unity property, since
+constants cannot be constructed otherwise.
+10
+
+Bernstein polynomials can be evaluated efficiently using the recursive formula
+bp
+0(ξ) = (1 − ξ)p ,
+bp
+i+1(ξ) =
+(p − i)ξ
+(p + 1)(1 − ξ)bp
+i (ξ) ,
+i ∈ {0, 1, ..., p − 1} ,
+(3.8)
+which allows for fast evaluation of the base functions.
+Remark 3.1
+Note that the formula Eq. (3.8) implies limξ→1 bp
+i+1(ξ) = ∞. As such, evaluations using the formula are required
+to use ξ < 1 preferably with additional tolerance. The limit case ξ = 1 is zero for all Bernstein base functions
+aside from the last function belonging to the vertex, which simply returns one
+bp
+i (1) = 0
+∀ i ̸= p ,
+bp
+p(1) = 1 .
+(3.9)
+3.2
+Dual numbers
+Dual numbers [16] can be used to define define an augmented algebra, where the derivative of a function can
+be computed simultaneously with the evaluation of the function. This enhancement is also commonly used
+in forward automatic differentiation [8, 37], not to be confused with numerical differentiation, since unlike in
+numerical differentiation, automatic differentiation is no approximation and yields the exact derivative. The
+latter represents an alternative method to finding the derivatives of base functions, as opposed to explicit
+formulas or approximations. Dual numbers augment the classical numbers by adding a non-zero number ε with
+a zero square ε2 = 0.
+Definition 3.1 (Dual number)
+The dual number is defined by
+x + x′ε ,
+ε ≪ 1 ,
+(3.10)
+where x′ is the derivative (only in automatic differentiation), ε is an abstract number (infinitesimal) and formally
+ε2 = 0.
+The augmented algebra results automatically from the definition of the dual number.
+Definition 3.2 (Augmented dual algebra)
+The standard algebraic operations take the following form for dual numbers
+1. Addition and subtraction
+(x + x′ε) ± (y + y′ε) = x ± y + (x′ ± y′)ε .
+(3.11)
+2. Multiplication
+(x + x′ε)(y + y′ε) = xy + (xy′ + x′y)ε ,
+(3.12)
+since formally ε2 = 0.
+3. Division is achieved by first defining the inverse element
+(x + x′ε)(y + y′ε) = 1
+⇐⇒
+y = 1
+x,
+y′ = − x′
+x2 ,
+(3.13)
+such that
+(x + x′ε)/(y + y′ε) = x/y + (x′/y − xy′/y2)ε .
+(3.14)
+Application of the above definitions to polynomials
+p(x + ε) =
+∞
+�
+i=0
+ci(x + ε)i =
+∞
+�
+i=0
+1
+�
+j=0
+ci
+�i
+j
+�
+xi−jεj =
+∞
+�
+i=0
+cixi + ε
+∞
+�
+i=1
+i cixi−1 = p(x) + p′(x)ε ,
+(3.15)
+allows the extension to various types of analytical functions with a power-series representation (such as trigono-
+metric or hyperbolic).
+11
+
+v1
+v3
+v2
+Γ
+ν
+τ
+ξ
+η
+x1
+x3
+x2
+Ae
+t
+n
+x
+y
+x : Γ → Ae
+Figure 4.1: Barycentric mapping of the reference triangle to an element in the physical domain.
+Definition 3.3 (General dual numbers function)
+A function of a dual number is defined in general by
+f(x + ε) = f(x) + f ′(x)ε ,
+(3.16)
+being a fundamental formula for forward automatic differentiation.
+The definition of dual numbers makes them directly applicable to the general rules of differentiation, such as
+the chain rule or product rule, in which case the derivative is simply the composition of previous computations
+with ε. The logic of dual numbers can be understood intuitively by the directional derivative
+d
+dxf(x) = ∂x′f(x) = d
+dεf(x + x′ε)
+����
+ε=0
+= lim
+ε→0
+f(x + x′ε) − f(x)
+ε
+,
+(3.17)
+where dividing by ε and setting ε = 0 are deferred to the last step of the computation, being the extraction of
+the derivative and equivalent to the operation f(x + ε) − f(x) with the augmented algebra of dual numbers.
+In this work we apply dual numbers for the computation of Bernstein polynomials using the recursive formula
+Eq. (3.8), thus allowing to iteratively compute each base function simultaneously with its derivative.
+4
+Triangular elements
+The triangle elements are mapped from the reference element Γ to the physical domain Ae via barycentric
+coordinates
+x(ξ, η) = (1 − ξ − η)x1 + η x2 + ξ x3 ,
+x : Γ → Ae ,
+Γ = {(ξ, η) ∈ [0, 1]2 | ξ + η ≤ 1} ,
+(4.1)
+where xi represent the coordinates of the vertices of one triangle in the physical domain, see Fig. 4.1. The
+corresponding Jacobi matrix reads
+J = Dx =
+�x3 − x1,
+x2 − x1
+�
+∈ R2×2 .
+(4.2)
+4.1
+The Bernstein-B´ezier basis for triangles
+The base functions on the triangle reference element are defined using the binomial expansion of the barycentric
+coordinates on the domain Γ
+1 = (λ1 + λ2 + λ3)p = ([1 − ξ − η] + η + ξ)p .
+(4.3)
+As such, the B´ezier base functions read
+bp
+ij(λ1, λ2, λ3) =
+�
+p
+i
+� �
+p − i
+j
+�
+λp−i−j
+1
+λj
+2λi
+3 ,
+(4.4)
+12
+
+(a)
+(b)
+(c)
+Figure 4.2: Cubic vertex (a), edge (b) and cell (c) B´ezier base functions on the reference triangle.
+(0,0)
+(1,0)
+(1,1)
+(0,1)
+α
+β
+Γ
+(0,0)
+(1,0)
+(0,1)
+ξ
+η
+ξ : α → Γ
+Figure 4.3: Duffy transformation from a quadrilateral to a triangle by collapse of the coordinate system.
+with the equivalent bivariate form
+bp
+ij(ξ, η) =
+�p
+i
+� �p − i
+j
+�
+(1 − ξ − η)p−i−jηjξi ,
+(4.5)
+of which some examples are depicted in Fig. 4.2. The Duffy transformation
+ξ : [0, 1]2 → Γ ,
+{α, β} �→ {ξ, η} ,
+(4.6)
+given by the relations
+ξ = α ,
+α = ξ ,
+η = (1 − α)β ,
+β =
+η
+1 − ξ ,
+(4.7)
+allows to view the triangle as a collapsed quadrilateral, see Fig. 4.3. Inserting the Duffy map into the definition
+of the B´ezier base function yields the split
+bp
+ij(ξ, η) =
+�p
+i
+� �p − i
+j
+�
+(1 − ξ − η)p−i−jηjξi
+=
+�p
+i
+� �p − i
+j
+�
+(1 − α − [1 − α]β)p−i−j(1 − α)jβjαi
+=
+�p
+i
+� �p − i
+j
+�
+(1 − α)p−i−j(1 − β)p−i−j(1 − α)jβjαi
+(4.8)
+=
+�p
+i
+�
+(1 − α)p−iαi
+�p − i
+j
+�
+(1 − β)p−i−jβj
+= bp
+i (α) bp−i
+j
+(β) .
+In other words, the Duffy transformation results in a natural factorization of the B´ezier triangle into Bernstein
+base functions [1]. The latter allows for fast evaluation using sum factorization. Further, it is now clear that
+B´ezier triangles are given by the interpolation of B´ezier curves, where the degree of the polynomial decreases
+13
+
+ξ
+η
+outer B´ezier curve with p = 3
+inner B´ezier curves with p < 3
+control polygon of η-curves
+outer B´ezier curves with p = 3
+inner B´ezier curves with p = 3
+Figure 4.4: B´ezier triangle built by interpolating B´ezier curves with an ever decreasing polynomial degree.
+v1
+v3
+v2
+ξ
+η
+Figure 4.5: Traversal order of base functions. The purple lines represent the order in which the base functions
+are constructed by the factorized evaluation. Note that the traversal order on each edge is intrinsically from
+the lower to the higher vertex index.
+between each curve, see Fig. 4.4. In order to compute gradients on the reference domain one applies the chain
+rule
+∇ξbp
+ij = (Dαξ)−T ∇αbp
+ij ,
+Dαξ =
+� 1
+0
+−β
+1 − α
+�
+,
+(Dαξ)−T =
+1
+1 − α
+�1 − α
+β
+0
+1
+�
+.
+(4.9)
+The factorization is naturally suited for the use of dual numbers since the α-gradient of a base function reads
+∇αbp
+ij(α, β) =
+�
+���
+bp−i
+j
+d
+dαbp
+i
+bp
+i
+d
+dβ bp−i
+j
+�
+��� ,
+(4.10)
+such that only the derivatives of the Bernstein base functions with respect to their parameter are required.
+The Duffy transformation induces an intrinsic optimal order of traversal of the base functions, compare
+Fig. 4.5, namely
+(i, j) = (0, 0) → (0, 1) → ... → (2, 2) → ... → (i, p − i) → ... → (p, 0) ,
+(4.11)
+which respects a clockwise orientation of the element, compare [52]. Thus, the order of the sequence of discrete
+values on common edges is determined by the global orientation. In order to relate a base function to a polytopal
+piece of the element, one observes the following result.
+Observation 4.1 (Triangle base functions)
+The polytope of each base function bp
+ij(ξ, η) can be determined as follows:
+14
+
+• The indices (0, 0), (0, p) and (p, 0) represent the first, second and last vertex base functions, respectively.
+• The indices (0, j) with 0 < j < p and (i, 0) with 0 < i < p represent the first and second edge base
+functions, respectively. Base functions of the slanted edge are given by (i, p − i) with 0 < i < p.
+• The remaining index combinations are cell base functions.
+With the latter observation, the construction of vertex-, edge- and cell base functions follows the intrinsic
+traversal order induced by the Duffy transformation and relates to a specific polytope via index-pairs.
+4.2
+N´ed´elec elements of the second type
+We construct the base functions for the N´ed´elec element of the second type using the polytopal template
+methodology introduced in [50]. The template sets read
+T1 = {e2, e1} ,
+T2 = {e1 + e2, e1} ,
+T3 = {e1 + e2, −e2} ,
+T12 = {e2, −e1} ,
+T13 = {e1, e2} ,
+T23 = {(1/2)(e1 − e2), e1 + e2} ,
+T123 = {e1, e2} .
+(4.12)
+The space of B´ezier polynomials is split across the polytopes of the reference triangle into
+Bp(Γ) =
+� 3
+�
+i=1
+Vp
+i (Γ)
+�
+⊕
+�
+�
+�
+�
+j∈J
+Ep
+j (Γ)
+�
+�
+� ⊕ Cp
+123(Γ) ,
+J = {(1, 2), (1, 3), (2, 3)} ,
+(4.13)
+where Vp
+i are the sets of the vertex base functions, Ep
+j are the sets of edge base functions, Cp
+123 is the set of cell
+base functions, and the ⊕ indicates summation over non-overlapping spaces. Consequently, the N´ed´elec basis
+is given by
+N p
+II =
+� 3
+�
+i=1
+Vp
+i ⊗ Ti
+�
+⊕
+�
+�
+�
+�
+j∈J
+Ep
+j ⊗ Tj
+�
+�
+� ⊕ {Cp
+123 ⊗ T123} ,
+J = {(1, 2), (1, 3), (2, 3)} .
+(4.14)
+Using the B´ezier basis one finds the following base functions, which inherit the optimal complexity of the
+underlying basis.
+Definition 4.1 (B´ezier-N´ed´elec II triangle basis)
+The following base functions are defined on the reference triangle.
+• On the edges the base function reads
+e12 :
+ϑ(ξ, η) = bp
+00e2 ,
+ϑ(ξ, η) = bp
+0p(e1 + e2) ,
+ϑ(ξ, η) = bp
+0je2 ,
+0 < j < p ,
+e13 :
+ϑ(ξ, η) = bp
+00e1 ,
+ϑ(ξ, η) = bp
+p0(e1 + e2) ,
+ϑ(ξ, η) = bp
+i0e1 ,
+0 < i < p ,
+e23 :
+ϑ(ξ, η) = bp
+0pe1 ,
+ϑ(ξ, η) = −bp
+p0e2 ,
+ϑ(ξ, η) = (1/2) bp
+i,p−i(e1 − e2) ,
+0 < i < p ,
+(4.15)
+where the first two base functions for each edge are the vertex-edge base functions and the third equation
+generates pure edge base functions.
+• The cell base functions read
+c123 :
+ϑ(ξ, η) = −bp
+0je1 ,
+0 < j < p ,
+ϑ(ξ, η) = bp
+i0e2 ,
+0 < i < p ,
+ϑ(ξ, η) = bp
+i,p−i(e1 + e2) ,
+0 < i < p ,
+ϑ(ξ, η) = bp
+ije2 ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η) = bp
+ije1 ,
+0 < i < p ,
+0 < j < p − i ,
+(4.16)
+15
+
+where the first three are the respective edge-cell base functions. The remaining two are pure cell base
+functions.
+4.3
+N´ed´elec elements of the first type
+In order to construct the N´ed´elec element of the first type we rely on the construction of the kernel introduced
+in [58] via the exact de Rham sequence and the polytopal template for the non-kernel base functions following
+[50]. The complete N´ed´elec space reads
+N p
+I = N 0
+I ⊕
+�
+�
+�
+�
+j∈J
+∇Ep+1
+j
+�
+�
+� ⊕ ∇Cp+1
+123 ⊕
+� 2
+�
+i=1
+Vp
+i ⊗ Ti
+�
+⊕
+�
+�
+�
+�
+j∈J
+Ep
+j ⊗ Tj
+�
+�
+� ⊕ {Cp
+123 ⊗ T123} ,
+J = {(1, 2), (1, 3), (2, 3)} ,
+(4.17)
+where we relied on the decomposition Eq. (4.14). Applying the construction to the B´ezier basis yields the
+following base functions.
+Definition 4.2 (B´ezier-N´ed´elec I triangle basis)
+We define the base functions on the reference triangle.
+• On the edges we employ the lowest order N´ed´elec base functions and the edge gradients
+e12 :
+ϑ(ξ, η) = ϑI
+1 ,
+ϑ(ξ, η) = ∇ξbp+1
+0j
+,
+0 < j < p + 1 ,
+e13 :
+ϑ(ξ, η) = ϑI
+2 ,
+ϑ(ξ, η) = ∇ξbp+1
+i0
+,
+0 < i < p + 1 ,
+e23 :
+ϑ(ξ, η) = ϑI
+3 ,
+ϑ(ξ, η) = ∇ξbp+1
+i,p+1−i ,
+0 < i < p + 1 .
+(4.18)
+• The cell functions read
+c123 :
+ϑ(ξ, η) = bp
+00ϑI
+3 ,
+ϑ(ξ, η) = bp
+0pϑI
+2 ,
+ϑ(ξ, η) = bp
+0j(ϑI
+3 − ϑI
+2) ,
+0 < j < p ,
+ϑ(ξ, η) = bp
+i0(ϑI
+1 + ϑI
+3) ,
+0 < i < p ,
+ϑ(ξ, η) = bp
+i,p−i(ϑI
+1 − ϑI
+2) ,
+0 < i < p ,
+ϑ(ξ, η) = bp
+ij(ϑI
+1 − ϑI
+2 + ϑI
+3) ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η) = ∇ξbp+1
+ij
+,
+0 < i < p + 1 ,
+0 < j < p + 1 − i ,
+(4.19)
+where the last formula gives the cell gradients and the remaining base functions are non-gradients.
+The definition relies on the base functions of the lowest order N´ed´elec element of the first type [5,50]
+ϑI
+1(ξ, η) =
+�
+η
+1 − ξ
+�
+,
+ϑI
+2(ξ, η) =
+�1 − η
+ξ
+�
+,
+ϑI
+3(ξ, η) =
+� η
+−ξ
+�
+.
+(4.20)
+5
+Tetrahedral elements
+The tetrahedral elements are mapped from the reference tetrahedron Ω by the three-dimensional barycentric
+coordinates onto the physical domain Ve, see Fig. 5.1
+x(ξ, η, ζ) = (1 − ξ − η − ζ)x1 + ζ x2 + η x3 + ξ x4 ,
+x : Ω → Ve ,
+Ω = {(ξ, η, ζ) ∈ [0, 1]3 | ξ + η + ζ ≤ 1} .
+(5.1)
+16
+
+ξ
+η
+ζ
+Ω
+v1
+v4
+v3
+v2
+τ
+ν
+Ve
+x2
+x1
+x3
+x4
+x
+y
+z
+t
+n
+x : Ω → Ve
+Figure 5.1: Barycentric mapping of the reference tetrahedron to an element in the physical domain.
+The corresponding Jacobi matrix reads
+J = Dx =
+�x4 − x1,
+x3 − x1,
+x2 − x1
+�
+∈ R3×3 .
+(5.2)
+5.1
+The Bernstein-B´ezier basis for tetrahedra
+Analogously to triangle elements, the B´ezier tetrahedra on the unit tetrahedron Ω are defined using the barycen-
+tric coordinates by expanding the coefficients of
+(λ1 + λ2 + λ3 + λ4)p = ([1 − ξ − η − ζ] + ζ + η + ξ)p = 1 ,
+(5.3)
+thus finding
+bp
+ijk(λ1, λ2, λ3, λ4) =
+�p
+i
+� �p − i
+j
+� �p − i − j
+k
+�
+λp−i−j−k
+1
+λk
+2λj
+3λk
+4 ,
+(5.4)
+with the equivalent trivariate form
+bp
+ijk(ξ, η, ζ) =
+�
+p
+i
+� �
+p − i
+j
+� �p − i − j
+k
+�
+(1 − ξ − η − ζ)p−i−j−kζkηjξi .
+(5.5)
+We construct the Duffy transformation by mapping the unit tetrahedron as a collapsed hexahedron
+ξ : [0, 1]3 → Ω ,
+{α, β, γ} �→ {ξ, η, ζ} ,
+(5.6)
+using the relations
+ξ = α ,
+η = (1 − α)β ,
+ζ = (1 − α)(1 − β)γ ,
+α = ξ ,
+β =
+η
+1 − ξ ,
+γ =
+ζ
+1 − ξ − η ,
+(5.7)
+as depicted in Fig. 5.2. Applying the Duffy transformation to B´ezier tetrahedra
+bp
+ijk(ξ, η, ζ) =
+�p
+i
+� �p − i
+j
+� �p − i − j
+k
+�
+(1 − ξ − η − ζ)p−i−j−kζkηjξi
+=
+�
+p
+i
+� �
+p − i
+j
+� �
+p − i − j
+k
+�
+(1 − α − (1 − α)β − (1 − α)(1 − β)γ)p−i−j−k
+· (1 − α)k(1 − β)kγk(1 − α)jβjαi
+=
+�p
+i
+� �p − i
+j
+� �p − i − j
+k
+�
+(1 − α)p−i−j−k(1 − β)p−i−j−k(1 − γ)p−i−j−k
+(5.8)
+· (1 − α)k(1 − β)kγk(1 − α)jβjαi
+=
+�p
+i
+�
+(1 − α)p−iαi
+�p − i
+j
+�
+(1 − β)p−i−jβj
+�p − i − j
+k
+�
+(1 − γ)p−i−j−kγk
+= bp
+i (α)bp−i
+j
+(β)bp−i−j
+k
+(γ) ,
+17
+
+α
+β
+γ
+(0,0,0)
+(1,0,0)
+(0,0,1)
+(1,1,0)
+(1,1,1)
+(0,1,1)
+ξ
+η
+ζ
+Ω
+(0,0,0)
+(1,0,0)
+(0,1,0)
+(0,0,1)
+ξ : α → Ω
+Figure 5.2: Duffy mapping of the unit hexahedron to the unit tetrahedron.
+leads to an intrinsic factorization via univariate Bernstein base functions, which allow for fast evaluations
+using sum factorization [1]. Further, since the pair bp−i
+j
+(β)bp−i−j
+k
+(γ) spans a B´ezier triangle, it is clear that
+the multiplication with bp
+i (α) interpolates between that triangle and a point in space, effectively spanning a
+tetrahedron. In order to compute gradients the chain rule is employed with respect to the Duffy transformation
+∇ξbp
+ijk = (Dαξ)−T ∇αbp
+ijk ,
+Dαξ =
+�
+�
+1
+0
+0
+−β
+1 − α
+0
+(β − 1)γ
+(α − 1)γ
+(1 − α)(1 − β)
+�
+� ,
+(Dαξ)−T =
+1
+(1 − α)(1 − β)
+�
+�
+(1 − α)(1 − β)
+(1 − β)β
+γ
+0
+1 − β
+γ
+0
+0
+1
+�
+� .
+(5.9)
+We use dual numbers to compute the derivative of each Bernstein base function and construct the α-gradient
+∇αbp
+ijk(α, β, γ) =
+�
+�������
+bp−i
+j
+bp−i−j
+k
+d
+dαbp
+i
+bp
+i bp−i−j
+k
+d
+dβ bp−i
+j
+bp
+i bp−i
+j
+d
+dγ bp−i−j
+k
+�
+�������
+.
+(5.10)
+The Duffy transformation results in the optimal order of traversal of the base functions depicted in Fig. 5.3.
+Note that the traversal order agrees with the oriental definitions introduced in [52] and each oriented face has
+the same order of traversal as the triangle Fig. 4.5. We relate the base functions to their respective polytopes
+using the index triplets.
+Observation 5.1 (Tetrahedron base functions)
+The polytope of each base function bp
+ijk(ξ, η, ζ) is determined as follows.
+• the indices (0, 0, 0), (0, 0, p), (0, p, 0) and (p, 0, 0) represent the respective vertex base functions;
+• the first edge is associated with the triplet (0, 0, k) where 0 < k < p, the second with (0, j, 0) where 0 < j < p
+and the third with (i, 0, 0) where 0 < i < p. The slated edges are given by (0, j, p − j) with 0 < j < p,
+(i, 0, p − i) with 0 < i < p and (i, p − i, 0) with 0 < i < p, respectively;
+• the base functions of the first face are given by (0, j, k) with 0 < j < p and 0 < k < p − j. The second face
+is associated with the base functions given by the triplets (i, 0, k) with 0 < i < p and 0 < k < p − i. The
+base functions of the third face are related to the indices (i, j, 0) with 0 < i < p and 0 < j < p − i. Lastly,
+the base functions of the slated face are given by (i, j, p − i − j) with 0 < i < p and 0 < j < p − i;
+• the remaining indices correspond to the cell base functions.
+Examples of B´ezier base functions on their respective polytopes are depicted in Fig. 5.4.
+18
+
+ξ
+η
+ζ
+v1
+v4
+v3
+v2
+Figure 5.3: Order of traversal of tetrahedral B´ezier base functions on the unit tetrahedron. The traversal order
+on each face agrees with an orientation of the vertices fijk = {vi, vj, vk} such that i < j < k. The traversal
+order on each edge is from the lower index vertex to the higher index vertex.
+(a)
+(b)
+(c)
+(d)
+Figure 5.4: Quartic B´ezier vertex (a), edge (b), face (c), and cell (c) base functions on the reference tetrahedron.
+19
+
+5.2
+N´ed´elec elements of the second type
+The B´ezier polynomial space is split according to the polytopes of the reference tetrahedron
+Bp(Ω) =
+� 4
+�
+i=1
+Vp
+i (Ω)
+�
+⊕
+�
+�
+�
+�
+j∈J
+Ep
+j (Ω)
+�
+�
+� ⊕
+��
+k∈K
+Fp
+k(Ω)
+�
+⊕ Cp
+1234(Ω) ,
+J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} ,
+K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} ,
+(5.11)
+where Vp
+i are the sets of vertex base functions, Ep
+j are the sets of edge base functions, Fp
+k are the sets of face
+base functions and Cp
+1234 is the set of cell base functions. We apply the template sets from [50]
+T1 = {e3, e2, e1} ,
+T2 = {e1 + e2 + e3, e2, e1} ,
+T3 = {e1 + e2 + e3, −e3, e1} ,
+T4 = {e1 + e2 + e3, −e3, −e2} ,
+T12 = {e3, −e2, −e1} ,
+T13 = {e2, e3, −e1} ,
+T14 = {e1, e3, e2} ,
+T23 = {e2, e1 + e2 + e3, −e1} ,
+T24 = {e1, e1 + e2 + e3, e2} ,
+T34 = {e1, e1 + e2 + e3, −e3} ,
+T123 = {e3, e2, −e1} ,
+T124 = {e3, e1, e2} ,
+T134 = {e2, e1, −e3} ,
+T234 = {e2, e1, e1 + e2 + e3} ,
+T1234 = {e3, e2, e1} ,
+(5.12)
+to span the N´ed´elec element of the second type
+N p
+II =
+� 4
+�
+i=1
+Vp
+i ⊗ Ti
+�
+⊕
+�
+�
+�
+�
+j∈J
+Ep
+j ⊗ Tj
+�
+�
+� ⊕
+��
+k∈K
+Fp
+k ⊗ Tk
+�
+⊕ {Cp
+1234 ⊗ T1234} ,
+J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} ,
+K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} .
+(5.13)
+We can now define the B´ezier-N´ed´elec element of the second type for arbitrary powers while inheriting optimal
+complexity.
+Definition 5.1 (B´ezier-N´ed´elec II tetrahedral basis)
+We define the base functions on the reference tetrahedron:
+• on the edges the base functions read
+e12 :
+ϑ(ξ, η, ζ) = bp
+000e3 ,
+ϑ(ξ, η, ζ) = bp
+00p(e1 + e2 + e3) ,
+ϑ(ξ, η, ζ) = bp
+00ke3 ,
+0 < k < p ,
+e13 :
+ϑ(ξ, η, ζ) = bp
+000e2 ,
+ϑ(ξ, η, ζ) = bp
+0p0(e1 + e2 + e3) ,
+ϑ(ξ, η, ζ) = bp
+0j0e2 ,
+0 < j < p ,
+e14 :
+ϑ(ξ, η, ζ) = bp
+000e1 ,
+ϑ(ξ, η, ζ) = bp
+p00(e1 + e2 + e3) ,
+ϑ(ξ, η, ζ) = bp
+i00e1 ,
+0 < i < p ,
+e23 :
+ϑ(ξ, η, ζ) = bp
+00pe2 ,
+ϑ(ξ, η, ζ) = −bp
+0p0e3 ,
+ϑ(ξ, η, ζ) = bp
+0j,p−je2 ,
+0 < j < p ,
+e24 :
+ϑ(ξ, η, ζ) = bp
+00pe1 ,
+ϑ(ξ, η, ζ) = −bp
+p00e3 ,
+ϑ(ξ, η, ζ) = bp
+i0,p−ie1 ,
+0 < i < p ,
+e34 :
+ϑ(ξ, η, ζ) = bp
+0p0e1 ,
+ϑ(ξ, η, ζ) = −bp
+p00e2 ,
+ϑ(ξ, η, ζ) = bp
+i,p−i,0e1 ,
+0 < i < p ,
+(5.14)
+where the first two base functions on each edge are the vertex-edge base functions;
+20
+
+• the face base functions are given by
+f123 :
+ϑ(ξ, η, ζ) = −bp
+00ke2 ,
+0 < k < p ,
+ϑ(ξ, η, ζ) = bp
+0j0e3 ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+0j,p−j(e1 + e2 + e3) ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+0jke3 ,
+0 < j < p ,
+0 < k < p − j ,
+ϑ(ξ, η, ζ) = bp
+0jke2 ,
+0 < j < p ,
+0 < k < p − j ,
+f124 :
+ϑ(ξ, η, ζ) = −bp
+00ke1 ,
+0 < k < p ,
+ϑ(ξ, η, ζ) = bp
+i00e3 ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+i0,p−i(e1 + e2 + e3) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+i0ke3 ,
+0 < i < p ,
+0 < k < p − i ,
+ϑ(ξ, η, ζ) = bp
+i0ke1 ,
+0 < i < p ,
+0 < k < p − i ,
+f134 :
+ϑ(ξ, η, ζ) = −bp
+0j0e1 ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+i00e2 ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+i,p−i,0(e1 + e2 + e3) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+ij0e2 ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η, ζ) = bp
+ij0e1 ,
+0 < i < p ,
+0 < j < p − i ,
+f234 :
+ϑ(ξ, η, ζ) = −bp
+0j,p−je1 ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+i0,p−ie2 ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = −bp
+i,p−i,0e3 ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+ij,p−i−je2 ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η, ζ) = bp
+ij,p−i−je1 ,
+0 < i < p ,
+0 < j < p − i ,
+(5.15)
+where the first three formulas for each face are the edge-face base functions;
+• finally, the cell base functions read
+c1234 :
+ϑ(ξ, η, ζ) = −bp
+0jke1 ,
+0 < j < p ,
+0 < k < p − j ,
+ϑ(ξ, η, ζ) = bp
+i0ke2 ,
+0 < i < p ,
+0 < k < p − i ,
+ϑ(ξ, η, ζ) = −bp
+ij0e3 ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η, ζ) = bp
+ij,p−i−j(e1 + e2 + e3) ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η, ζ) = bp
+ijke3 ,
+0 < i < p ,
+0 < j < p − i ,
+0 < k < p − i − j ,
+ϑ(ξ, η, ζ) = bp
+ijke2 ,
+0 < i < p ,
+0 < j < p − i ,
+0 < k < p − i − j ,
+ϑ(ξ, η, ζ) = bp
+ijke1 ,
+0 < i < p ,
+0 < j < p − i ,
+0 < k < p − i − j ,
+(5.16)
+where the first four formulas are the face-cell base functions.
+5.3
+N´ed´elec elements of the first type
+In order to construct the N´ed´elec element of first type on tetrahedra we introduce the template sets
+T1 = {ϑI
+4, ϑI
+5, ϑI
+6} ,
+T2 = {−ϑI
+2, −ϑI
+3, ϑI
+6} ,
+T3 = {−ϑI
+3, −ϑI
+5} ,
+T12 = {ϑI
+4 − ϑI
+2, ϑI
+5 − ϑI
+3} ,
+T13 = {ϑI
+1 + ϑI
+4, ϑI
+6 − ϑI
+3} ,
+T14 = {ϑI
+1 + ϑI
+5, ϑI
+2 + ϑI
+6} ,
+T23 = {ϑI
+1 − ϑI
+2, ϑI
+6 − ϑI
+5} ,
+T24 = {ϑI
+1 − ϑI
+3, ϑI
+4 + ϑI
+6} ,
+T34 = {ϑI
+2 − ϑI
+3, ϑI
+4 − ϑI
+5} ,
+T123 = {ϑI
+1 − ϑI
+2 + ϑI
+4} ,
+T124 = {ϑI
+1 − ϑI
+3 + ϑI
+5} ,
+T134 = {ϑI
+2 − ϑI
+3 + ϑI
+6} ,
+T234 = {ϑI
+4 − ϑI
+5 + ϑI
+6} ,
+(5.17)
+21
+
+which are based on the lowest order N´ed´elec base functions on the unit tetrahedron
+ϑ1(ξ, η, ζ) =
+�
+�
+ζ
+ζ
+1 − ξ − η
+�
+� ,
+ϑ2(ξ, η, ζ) =
+�
+�
+η
+1 − ξ − ζ
+η
+�
+� ,
+ϑ3(ξ, η, ζ) =
+�
+�
+1 − η − ζ
+ξ
+ξ
+�
+� ,
+ϑ4(ξ, η, ζ) =
+�
+�
+0
+ζ
+−η
+�
+� ,
+ϑ5(ξ, η, ζ) =
+�
+�
+ζ
+0
+−ξ
+�
+� ,
+ϑ6(ξ, η, ζ) =
+�
+�
+η
+−ξ
+0
+�
+� .
+(5.18)
+For the non-gradient cell functions we use the construction introduced in [2]
+Rp =
+�
+(p + 1)bp
+i−ej∇λj −
+ij
+p + 1∇ξbp+1
+i
+| i ∈ Io
+�
+,
+(5.19)
+where Io is the set of multi-indices of cell functions, ej is the unit multi-index with the value one at position
+j and ij is the value of the i-multi-index at position j. Note that only the first term in the cell functions is
+required to span the next space in the sequence due to
+curl
+�
+[p + 1]bp
+i−ej∇ξλj −
+ij
+p + 1∇ξbp+1
+i
+�
+= curl([p + 1]bp
+i−ej∇ξλj) .
+(5.20)
+However, without the added gradient the function would not belong to [Pp]3 ⊕ξ ×[�P]3 and consequently, would
+not be part of the N´ed´elec space. By limiting Rp to Rp
+∗ such that Rp
+∗ contains only the surface permutations
+with ∇λj = ej and the cell permutations with j ∈ {1, 2}, one retrieves the necessary base functions. The
+sum of the lowest order N´ed´elec base functions, the template base functions, gradient base functions, and the
+non-gradient cell base functions yields exactly (p+4)(p+3)(p+1)/2, thus satisfying the required dimensionality
+of the N´ed´elec space. The complete space reads
+N p
+I = N 0
+I ⊕
+��
+i∈I
+∇Ep+1
+i
+�
+⊕
+�
+�
+�
+�
+j∈J
+∇Fp+1
+j
+�
+�
+� ⊕ ∇Cp+1
+1234 ⊕
+� 3
+�
+k=1
+Vp
+k ⊗ Tk
+�
+⊕
+��
+i∈I
+Ep
+i ⊗ Ti
+�
+⊕
+�
+�
+�
+�
+j∈J
+Fp
+j ⊗ Tj
+�
+�
+� ⊕ Rp+1
+∗
+,
+I = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} ,
+J = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)}
+.
+(5.21)
+Here, the B´ezier basis is used to construct the higher order N´ed´elec base functions of the first type.
+Definition 5.2 (B´ezier-N´ed´elec I tetrahedral basis)
+The base functions are defined on the reference tetrahedron:
+• for the edges we use the lowest order base functions from Eq. (5.18). The remaining edge base functions
+are given by the gradients
+e12 :
+ϑ(ξ, η, ζ) = ϑI
+1 ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+00k ,
+0 < k < p + 1 ,
+e13 :
+ϑ(ξ, η, ζ) = ϑI
+2 ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+0j0 ,
+0 < j < p + 1 ,
+e14 :
+ϑ(ξ, η, ζ) = ϑI
+3 ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+i00 ,
+0 < i < p + 1 ,
+e23 :
+ϑ(ξ, η, ζ) = ϑI
+4 ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+0j,p+1−j ,
+0 < j < p + 1 ,
+e24 :
+ϑ(ξ, η, ζ) = ϑI
+5 ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+i0,p+1−i ,
+0 < i < p + 1 ,
+e34 :
+ϑ(ξ, η, ζ) = ϑI
+6 ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+00k ,
+0 < i < p + 1 ;
+(5.22)
+22
+
+• on faces we employ both template base functions and gradients
+f123 :
+ϑ(ξ, η, ζ) = bp
+000ϑI
+4 ,
+ϑ(ξ, η, ζ) = −bp
+00pϑI
+2 ,
+ϑ(ξ, η, ζ) = bp
+00k(ϑI
+4 − ϑI
+2) ,
+0 < k < p ,
+ϑ(ξ, η, ζ) = bp
+0j0(ϑI
+1 + ϑI
+4) ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+0j,p−j(ϑI
+1 − ϑI
+2) ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+0jk(ϑI
+1 − ϑI
+2 + ϑI
+4) ,
+0 < j < p ,
+0 < k < p − j ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+0jk ,
+0 < j < p + 1 ,
+0 < k < p + 1 − j ,
+f124 :
+ϑ(ξ, η, ζ) = bp
+000ϑI
+5 ,
+ϑ(ξ, η, ζ) = −bp
+00pϑI
+3 ,
+ϑ(ξ, η, ζ) = bp
+00k(ϑI
+5 − ϑI
+3) ,
+0 < k < p ,
+ϑ(ξ, η, ζ) = bp
+i00(ϑI
+1 + ϑI
+5) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+i0,p−i(ϑI
+1 − ϑI
+3) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+i0k(ϑI
+1 − ϑI
+3 + ϑI
+5) ,
+0 < i < p ,
+0 < k < p − i ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+i0k ,
+0 < i < p + 1 ,
+0 < k < p + 1 − i ,
+f134 :
+ϑ(ξ, η, ζ) = bp
+000ϑI
+6 ,
+ϑ(ξ, η, ζ) = −bp
+0p0ϑI
+3 ,
+ϑ(ξ, η, ζ) = bp
+0j0(ϑI
+6 − ϑI
+3) ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+i00(ϑI
+2 + ϑI
+6) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+i,p−i,0(ϑI
+2 − ϑI
+3) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+ij0(ϑI
+2 − ϑI
+3 + ϑI
+6) ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+ij0 ,
+0 < i < p + 1 ,
+0 < j < p + 1 − i ,
+f234 :
+ϑ(ξ, η, ζ) = bp
+00pϑI
+6 ,
+ϑ(ξ, η, ζ) = −bp
+0p0ϑI
+5 ,
+ϑ(ξ, η, ζ) = bp
+0j,p−j(ϑI
+6 − ϑI
+5) ,
+0 < j < p ,
+ϑ(ξ, η, ζ) = bp
+i0,p−i(ϑI
+4 + ϑI
+6) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+i,p−i,0(ϑI
+4 − ϑI
+5) ,
+0 < i < p ,
+ϑ(ξ, η, ζ) = bp
+ij,p−i−j(ϑI
+4 − ϑI
+5 + ϑI
+6) ,
+0 < i < p ,
+0 < j < p − i ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+ij,p−i.j ,
+0 < i < p + 1 ,
+0 < j < p + 1 − i ;
+(5.23)
+• the cell base functions read
+c1234 :
+ϑ(ξ, η, ζ) = (p + 2)bp+1
+i−1,jke1 −
+i
+p + 2∇ξbp+2
+ijk ,
+0 < i < p + 2 ,
+0 < j < p + 2 − i ,
+0 < k < p + 2 − i − j
+,
+ϑ(ξ, η, ζ) = (p + 2)bp+1
+i,j−1,ke2 −
+j
+p + 2∇ξbp+2
+ijk ,
+0 < i < p + 2 ,
+0 < j < p + 2 − i ,
+0 < k < p + 2 − i − j
+,
+ϑ(ξ, η, ζ) = (p + 2)bp+1
+ij0 e3 −
+1
+p + 2∇ξbp+2
+ij1 ,
+0 < i < p + 2 ,
+0 < j < p + 2 − i ,
+ϑ(ξ, η, ζ) = ∇ξbp+1
+ijk ,
+0 < i < p + 1 ,
+0 < j < p + 1 − i ,
+0 < k < p + 1 − i − j
+.
+(5.24)
+23
+
+6
+Numerical quadrature
+Although the base functions are expressed using (α, β, γ) the domain is either the reference triangle or the
+reference tetrahedron, which require fewer quadrature points than their counterparts given by the Duffy trans-
+formation (quad or hexahedron). As such, we employ a mixture of the efficient quadrature points introduced
+in [14,19,39,56,57] for triangles and tetrahedra, where we avoid quadrature schemes with points on the edges
+or faces of the reference domain due to the recursion formula of the Bernstein polynomials Eq. (3.8). The
+quadrature points are mapped to their equivalent expression in (α, β, γ). Consequently, the integration over the
+reference triangle or tetrahedron reads
+�
+Ae
+f(x, y) dA =
+�
+Γ
+(f ◦ (ξ, η))(α, β) | det J| dΓ ,
+�
+Ve
+f(x, y, z) dV =
+�
+Ω
+(f ◦ (ξ, η, ζ))(α, β, γ) | det J| dΩ .
+(6.1)
+For the lower order elements we use the Lagrangian-N´ed´elec construction from [52,53].
+7
+Boundary conditions
+The degrees of freedom in [12] commute between the continuous and discrete spaces.
+As such, they allow
+to exactly satisfy the consistent coupling condition [11].
+We note that the functionals can be viewed as a
+hierarchical system of Dirichlet boundary problems. In the case of hierarchical base functions [58], they can
+be solved independently. However, here the boundary value of each polytope is required in advance due to the
+non-hierarchical nature of Bernstein polynomials. In other words, one must first solve the problem for vertices,
+then for edges, afterwards for faces, and finally for the cell. In our case the degrees of freedom for the cell are
+irrelevant since a cell is never part of the boundary.
+7.1
+Boundary vertices
+The finite element mesh identifies each vertex with a tuple of coordinates. It suffices to evaluate the displacement
+field at the vertex
+ud
+i = �u
+����
+xi
+.
+(7.1)
+If the field is vectorial, each component is evaluated at the designated vertex. The boundary conditions of the
+microdistortion field are associated with tangential projections and as such do not have vertex-type degrees of
+freedom. This is the case since a vertex does not define a unique tangential plane.
+7.2
+Boundary edges
+The edge functionals from [12] for the H 1-conforming subspace
+lij(u) =
+�
+si
+∂qj
+∂s
+∂u
+∂s ds ,
+q ∈ Pp(s) ,
+(7.2)
+can be reformulated for a reference edge on a unit domain α ∈ [0, 1]. We parametrize the edge via
+x(α) = (1 − α)x1 + αx2 .
+(7.3)
+As such, the following relation exists between the unit parameter and the arc-length parameter
+t = d
+dαx = x2 − x1 ,
+ds = ∥dx∥ = ∥x2 − x1∥dα = ∥t∥dα .
+(7.4)
+24
+
+α
+0
+1
+ξ : α → Γ
+ξ2
+ξ1
+Γ
+τ
+ξ
+η
+x2
+x1
+A
+t
+x
+y
+x : Γ → A
+Figure 7.1: Barycentric mapping of edges from the unit domain to the reference triangle and onto the physical
+domain.
+By the chain rule we find
+du
+ds = du
+dα
+dα
+ds = ∥t∥−1 du
+dα ,
+(7.5)
+for some function u. On edges, the test and trial functions are Bernstein polynomials parametrized by the unit
+domain. The function representing the boundary condition �u(x) however, is parametrized by the Cartesian
+coordinates of the physical space. We find its derivative with respect to the arc-length parameter by observing
+d
+ds �u = ⟨ d
+dsx, ∇x�u⟩ .
+(7.6)
+The derivative of the coordinates with respect to the arc-length is simply the normed tangent vector
+d
+dsx = dx
+dα
+dα
+ds = ∥t∥−1t .
+(7.7)
+Consequently, the edge boundary condition is given by
+�
+si
+∂qj
+∂s
+∂u
+∂s ds =
+� 1
+0
+�
+∥t∥−1 dqj
+dα
+� �
+∥t∥−1 du
+dα
+�
+∥t∥ dα
+=
+� 1
+0
+�
+∥t∥−1 dqj
+dα
+�
+⟨∥t∥−1t, ∇x�u⟩∥t∥ dα =
+�
+si
+∂qj
+∂s
+∂�u
+∂s ds
+∀ qj ∈ Pp(α) ,
+(7.8)
+and can be solved by assembling the stiffness matrix of the edge and the load vector induced by the prescribed
+displacement field �u, representing volume forces
+kij =
+� 1
+0
+�
+∥t∥−1 dni
+dα
+� �
+∥t∥−1 dnj
+dα
+�
+∥t∥ dα ,
+fi =
+� 1
+0
+⟨∥t∥−1t, ∇x�u⟩
+�
+∥t∥−1 dni
+dα
+�
+∥t∥ dα .
+(7.9)
+Next we consider the Dirichlet boundary conditions for the microdistortion with the N´ed´elec space of the
+second type NII. The problem reads
+�
+si
+qj⟨t, p⟩ ds =
+�
+si
+qj⟨t, ∇x�u⟩ ds
+∀ qj ∈ Pp(si) .
+(7.10)
+Observe that on the edge the test functions qj are chosen to be the Bernstein polynomials. Further, by the
+polytopal template construction of the NII-space there holds ⟨t, θi⟩|s = ni(α). Therefore, the components of
+the corresponding stiffness matrix and load vectors read
+kij =
+� 1
+0
+ni nj∥t∥ dα ,
+fi =
+� 1
+0
+ni⟨t, ∇x�u⟩∥t∥ dα .
+(7.11)
+25
+
+Note that in order to maintain the exactness property, the degree of the N´ed´elec spaces N p
+I , N p
+II is always one
+less than the degree of the subspace Bp+1.
+Lastly, we consider the N´ed´elec element of the first type. The problem is given by
+�
+si
+qj⟨t, p⟩ ds =
+�
+si
+qj⟨t, ∇x�u⟩ ds
+∀ qj ∈ Pp(si) .
+(7.12)
+We define
+qi = d
+dαnp+1
+i
+,
+(7.13)
+and observe that on the edges the N´ed´elec base functions yield
+⟨t, θj⟩ = ⟨t, ∇xnp+1
+j
+⟩ = d
+dαnp+1
+j
+.
+(7.14)
+Therefore, the components of the stiffness matrix and the load vector result in
+kij =
+� 1
+0
+dnp+1
+i
+dα
+dnp+1
+j
+dα
+∥t∥ dα ,
+fi =
+� 1
+0
+dnp+1
+i
+dα
+⟨t, ∇x�u⟩∥t∥ dα .
+(7.15)
+7.3
+Boundary faces
+We start with the face boundary condition for the H 1-conforming subspace. The problem reads
+�
+Ai
+⟨∇fqj, ∇fu⟩ dA =
+�
+Ai
+⟨∇fqj, ∇f �u⟩ dA
+∀ qj ∈ Pp(Ai) .
+(7.16)
+The surface is parameterized by the barycentric mapping from the unit triangle Γ = {(ξ, η) ∈ [0, 1]2 | ξ +η ≤ 1}.
+The surface gradient is given by
+∇f �u = ∇x�u −
+1
+∥n∥2 ⟨∇x�u, n⟩n ,
+(7.17)
+where n is the surface normal. The surface gradient can also be expressed via
+∇fu = ei∂x
+i u = gβ∂ξ
+βu ,
+β ∈ {1, 2} ,
+(7.18)
+where ∂x
+β are partial derivates with respect to the physical coordinates, ∂ξ
+β are partial derivatives with respect
+to the reference domain and gβ are the contravariant base vectors. The Einstein summation convention over
+corresponding indices is implied. The covariant base vectors are given by
+gβ = ∂x
+∂ξβ .
+(7.19)
+One can find the contravariant vector orthogonal to the surface by
+g3 = n = g1 × g2 .
+(7.20)
+We define the mixed transformation matrix
+T =
+�
+g1 , g2 , g3�
+.
+(7.21)
+Due to the orthogonality relation ⟨gi, gj⟩ = δ j
+i the transposed inverse of T is clearly
+T −T =
+�
+g1 , g2 , g3
+�
+.
+(7.22)
+Thus, we can compute the surface gradient of functions parametrized by the reference triangle via
+∇fu =
+�
+g1 , g2�
+∇ξu = T −T
+∗
+∇ξu ,
+T −T
+∗
+=
+�
+g1 , g2�
+.
+(7.23)
+26
+
+Further, there holds the following relation between the physical surface and the reference surface
+dA = ∥n∥dΓ = ∥g3∥dΓ =
+�
+⟨g1 × g2, g3⟩ dΓ =
+√
+det T dΓ .
+(7.24)
+Consequently, we can write the components of the stiffness matrix and load vector as
+kij =
+�
+Γ
+⟨T −T
+∗
+∇ξni, T −T
+∗
+∇ξnj⟩
+√
+det T dΓ ,
+fi =
+�
+Γ
+⟨T −T
+∗
+∇ξni, ∇x�u − (det T )−1⟨∇x�u, n⟩n⟩
+√
+det T dΓ =
+�
+Γ
+⟨T −T
+∗
+∇ξni, ∇x�u⟩
+√
+det T dΓ ,
+(7.25)
+with the orthogonality ⟨gβ, n⟩ = 0 for β ∈ {1, 2}.
+In order to embed the consistent coupling boundary condition to the microdistortion we deviate from the
+degrees of freedom defined in [12] and apply the simpler H (divR)-projection
+⟨qi, p, ⟩H(divR) = ⟨qi, ∇f �u⟩H(divR)
+∀ qi ∈ N p
+I (A)
+or
+∀ qi ∈ N p
+II(A) .
+(7.26)
+Due to ker(curl) = ∇H 1 the problem reduces to
+�
+Ai
+⟨qj, p⟩ + ⟨curl2Dqj, curl2Dp⟩ dA =
+�
+Ai
+⟨qj, ∇f �u⟩ dA
+∀ qj ∈ N p
+I (A)
+or
+∀ qj ∈ N p
+II(A) .
+(7.27)
+We express the co- and contravariant Piola transformation from the two-dimensional reference domain to the
+three-dimensional physical domain using
+θi = T −T
+∗
+ϑi ,
+divx R θi =
+1
+√
+det T
+divξ R ϑi .
+(7.28)
+Thus, the stiffness matrix components and load vector components read
+kij =
+�
+Γ
+⟨T −T
+∗
+ϑi, T −T
+∗
+ϑj⟩ + ⟨(det T )−1/2 divξ R ϑi, (det T )−1/2 divξ R ϑj⟩
+√
+det T dΓ ,
+fi =
+�
+Γ
+⟨T −T
+∗
+ϑi, ∇x�u − (det T )−1⟨∇x�u, n⟩n⟩
+√
+det T dΓ =
+�
+Γ
+��T −T
+∗
+ϑi, ∇x�u⟩
+√
+det T dΓ ,
+(7.29)
+where we again make use of the orthogonality between the surface tangent vectors and its normal vector.
+8
+Numerical examples
+In the following we test the finite element formulations with an artificial analytical solution in the antiplane shear
+model and with an analytical solution for an infinite plane under cylindrical bending in the three dimensional
+model. Finally, we benchmark the ability of the finite element formulations to correctly interpolate between
+micro Cmicro and macro Cmacro stiffnesses as described by the characteristic length scale parameter Lc. The
+majority of convergence results are presented by measuring the error in the Lebesgue norm over the domain
+∥�u − uh∥L2 =
+��
+V
+∥�u − uh∥2 dV ,
+∥ �P − P h∥L2 =
+��
+V
+∥ �P − P h∥2 dV ,
+(8.1)
+in which context {�u, �P } and {uh, P h} are the analytical and approximate subspace solutions, respectively.
+8.1
+Compatible microdistortion
+In [53] we explored the conditions for which the microdistortion p reduces to a gradient field, i.e. p is compatible.
+By defining the micro-moment with a scalar potential
+m = ∇100 − x2 − y2
+10
+= −1
+5
+�x
+y
+�
+,
+(8.2)
+27
+
+and constructing an analytical solution for the displacement field
+�u = sin
+�x2 + y2
+5
+�
+,
+(8.3)
+we can recover the analytical solution of the microdistortion
+p =
+1
+µe + µmicro
+(m + µe∇�u) = 1
+2
+�
+−1
+5
+�x
+y
+�
++ 2
+5
+�x cos([x2 + y2]/5)
+y cos([x2 + y2]/5)
+��
+= 1
+5
+�x cos([x2 + y2]/5)
+y cos([x2 + y2]/5)
+�
+− 1
+10
+�x
+y
+�
+,
+(8.4)
+where for simplicity we set all material constants to one. Since m is a gradient field, the microdistortion p is
+also reduced to a gradient field and curl2Dp = 0, see [53]. Note that this result is specific to antiplane shear
+and does not generalize to the full three-dimensional model, compare [52]. We note that the microdistortion
+is not equal to the gradient of the displacement field and as such, their tangential projections on an arbitrary
+boundary are not automatically the same. However, for both the gradient of the displacement field and the
+micro-moment is the tangential projection on the boundary of the circular domain A = {x ∈ R2 | ∥x∥ ≤ 10}
+equal to zero
+⟨∇t, �u⟩
+����
+∂A
+= ⟨t, m⟩
+����
+∂A
+= 0 ,
+(8.5)
+and as such the microdistortion belongs to p ∈ H0(curl, A).
+Consequently, we can set sD = ∂A and the
+consistent coupling condition remains compatible.
+With the displacement and the microdistortion fields at
+hand we derive the corresponding forces
+f = 1
+25
+�
+2x2 sin
+�x2 + y2
+5
+�
++ 2y2 sin
+�x2 + y2
+5
+�
+− 10 cos
+�x2 + y2
+5
+�
+− 5
+�
+.
+(8.6)
+The approximation of the displacement and microdistortion fields using linear and higher order elements is
+shown in Fig. 8.1. We note that even with almost 3000 finite elements and 6000 degrees of freedom the linear
+formulation is incapable of finding an adequate approximation. On the other side of the spectrum, the higher
+order approximation (degree 7) with 57 elements and 4097 degrees of freedom yields very accurate results in
+the interior of the domain. However, the exterior of the domain is captured rather poorly. This is the case since
+the geometry of the circular domain is being approximated by linear triangles. Thus, in this setting, a finer
+mesh captures the geometry in a more precise manner. The effects of the geometry on the approximation of the
+solution are also clearly visible in the convergence graphs in Fig. 8.2; only after a certain accuracy in the domain
+description is achieved do the finite elements retrieve their predicted convergence rates, compare [52,53]. This
+is clearly observable when comparing the convergence curves of the linear and seventh order elements. The
+linear element generates quadratic convergence p + 1 = 1 + 1 = 2, whereas the seventh-order element yields
+the convergence slope 7 (where 8 is expected).
+Although the seventh-order formulation encompasses more
+degrees of freedom, it employs a coarser mesh and as such, generates higher errors at the boundary. The errors
+themselves can be traced back to the consistent coupling condition since, for a non-perfect circle the gradient
+of the displacement field induces tangential projections on the imperfect boundary. The influence of the latter
+effect is even more apparent in the convergence of the microdistortion, where the higher order formulations are
+unable to perform optimally on coarse meshes.
+8.2
+Cylindrical bending
+In order to test the capability of the finite element formulations to capture the intrinsic behaviour of the relaxed
+micromorphic model, we compare with analytical solutions of boundary-value problems. The first example
+considers the displacement and microdistortion fields under cylindrical bending [43] for infinitely extended
+plates. Let the plates be defined as V = (−∞, ∞)2 × [−1/2, 1/2], than the analytical solution for cylindrical
+bending reads
+u = κ
+�
+�
+−xz
+0
+x2/2
+�
+� ,
+P = −κ
+�
+�
+[41z + 20
+√
+82 sech(
+�
+41/2) sinh(
+√
+82z)]/1681
+0
+x
+0
+0
+0
+−x
+0
+0
+�
+� ,
+(8.7)
+28
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+(j)
+(k)
+(l)
+Figure 8.1: Depiction of the displacement field (a)-(c) and the microdistortion field (d)-(f) for the antiplane
+shear problem, for the linear element under h-refinement with 225, 763 and 2966 elements, corresponding to
+485, 1591 and 6060 degrees of freedom. The p-refinement of the displacement field on the coarsest mesh of 57
+elements is visualized in (g)-(l) with p ∈ {3, 5, 7}, corresponding to 731, 2072 and 4097 degrees of freedom.
+29
+
+11
+NA144
+44103
+104
+105
+10−3
+10−1
+101
+degrees of freedom
+∥�u − uh∥L2
+L1 × N 0
+I
+L2 × N 1
+II
+B3 × N 2
+II
+B5 × N 4
+II
+B7 × N 6
+II
+O(h2)
+O(h7)
+(a)
+103
+104
+105
+10−3
+10−1
+101
+degrees of freedom
+∥�p − ph∥L2
+L1 × N 0
+I
+L2 × N 1
+II
+B3 × N 2
+II
+B5 × N 4
+II
+B7 × N 6
+II
+O(h)
+O(h2)
+(b)
+Figure 8.2: Convergence of displacement (a) and the microdistortion (b) under h-refinement for multiple poly-
+nomial degrees for the antiplane shear problem.
+where sech(x) = 1/ cosh(x), and for the following values of material constants
+λe = λmicro = 0 ,
+µe = µmacro = 1/2 ,
+µc = 0 ,
+Lc = 1 ,
+µmicro = 20 .
+(8.8)
+The intensity of the curvature parameter κ of the plate is chosen to be κ = 14/200.
+Remark 8.1
+The particular case of the cylindrical bending for which λe = λmicro = 0 (equivalent to a zero micro-Poisson’s
+ratio) has been solved, along with its more general case (λe ̸= λmicro ̸= 0), in [43]. The advantage of considering
+this particular case is that a cut out finite plate of the infinite domain automatically exhibits the consistent
+coupling boundary conditions on its side surfaces.
+Remark 8.2
+Note that the general analytical solution for cylindrical bending does not depend on µc, so we can set µc = 0
+without loss of generality, compare [43].
+We define the finite domain V = [−10, 10]2 × [−1/2, 1/2] and the boundaries
+AD1 = {−10} × [−10, 10] × [−1/2, 1/2] ,
+AD2 = {10} × [−10, 10] × [−1/2, 1/2] ,
+AN = ∂V \ {AD1 ⊕ AD2} .
+(8.9)
+Additionally, on the Dirichlet boundary we impose the translated analytical solution �u = u −
+�0
+0
+3.5�T .
+The displacement field and the last row of the microdistortion are depicted in Fig. 8.3. The displacement
+field is dominated by its quadratic term and captured correctly.
+The last row of the microdistortion is a
+linear function and easily approximated even with linear elements. On the contrary, the P11 component of
+the microdistortion is a hyperbolic function of the z-axis.
+The results of its approximation at x = y = 0
+(the centre of the plane) are given in Fig. 8.4. We observe that even increasing the number of linear finite
+elements to the extreme only results in better oscillations around the analytical solution. In comparison, higher
+order formulations converge towards the expected hyperbolic behaviour. The approximation of the quadratic
+N´ed´elec element of the first type is nearly perfect, whereas its second type counterpart clearly deviates from
+the analytical solution at z ≈ −0.25. Taking the cubic second type element yields the desired result. This
+phenomenon is an evident indicator of the prominent role of the Curl of the microdistortion in this type of
+problems. Firstly, the microdistortion is a non-gradient field. Secondly, the Curl of the analytical solution
+induces an hyperbolic sine term. Such functions are often approximated using at least cubic terms in power
+series, thus explaining the necessity of such high order elements for correct computations.
+30
+
+(a)
+(b)
+Figure 8.3: Displacement (a) and last row of the microdistortion (b) for the quadratic formulation using the
+N´ed´elec element of the first type.
+−0.5
+0
+0.5
+−1
+0
+1
+·10−2
+z-axis
+P11(z)
+ne = 5640
+ne = 44592
+ne = 354720
+(a)
+−0.5
+0
+0.5
+−1
+0
+1
+·10−2
+z-axis
+P11(z)
+B2 × N 1
+I
+B3 × N 2
+I
+L2 × N 1
+II
+B3 × N 2
+II
+B4 × N 3
+II
+(b)
+Figure 8.4: Convergence of the lowest order formulation under h-refinement with 732, 5640 and 44592 elements
+(a) and of the higher order formulations under p-refinement using 732 elements(b) towards the analytical solution
+(dashed curve) of the P11(z) component at x = y = 0.
+31
+
+8.3
+Bounded stiffness property
+The characteristic length scale parameter Lc allows the relaxed micromorphic model to capture the transition
+from highly homogeneous materials to materials with a pronounced micro-structure by governing the influence
+of the micro-structure on the overall behaviour of the model. We demonstrate this property of the model with
+an example, where we vary Lc and measure the resulting energy.
+Let the domain be given by the axis-symmetric cube V = [−1, 1]3 with a total Dirichlet boundary
+AD1 = {(x, y, z) ∈ [−1, 1]3 | x = ±1} ,
+AD2 = {(x, y, z) ∈ [−1, 1]3 | y = ±1} ,
+AD3 = {(x, y, z) ∈ [−1, 1]3 | z = ±1} ,
+(8.10)
+we embed the periodic boundary conditions
+�u
+����
+AD1
+=
+�
+�
+(1 − y2) sin(π[1 − z2])/10
+0
+0
+�
+� ,
+�u
+����
+AD2
+=
+�
+�
+0
+(1 − x2) sin(π[1 − z2])/10
+0
+�
+� ,
+�u
+����
+AD3
+=
+�
+�
+0
+0
+(1 − y2) sin(π[1 − x2])/10
+�
+� .
+(8.11)
+The material parameters are chosen as
+λmacro = 2 ,
+µmacro = 1 ,
+λmicro = 10 ,
+µmicro = 5 ,
+µc = 1 ,
+(8.12)
+thus giving rise to the following meso-parameters via Eq. (2.19)
+λe = 2.5 ,
+µe = 1.25 .
+(8.13)
+The displacement field as well as some examples of the employed meshes are shown in Fig. 8.5. In order to
+compute the upper and lower bound on the energy we utilize the equivalent Cauchy model formulation with
+the micro- and macro elasticity parameters. In order to assert the high accuracy of the solution of the bounds
+we employ tenth order finite elements. The progression of the energy in dependence of the characteristic length
+parameter Lc is given in Fig. 8.6. We observe the high mesh dependency of the lower order formulations, where
+the energy is clearly overestimated. The higher order formulations all capture the upper bound correctly but
+diverge with respect to the result of the lower bound. Notably, the approximation using the N´ed´elec element
+of the first type is more accurate than the equivalent formulation with the N´ed´elec element of the second type,
+thus indicating the non-negligible involvement of the micro-dislocation in the energy. Using standard mesh
+coarseness the cubic element formulation with N´ed´elec elements of the first type yields satisfactory results. In
+order to achieve the same on highly coarse meshes, one needs to employ seventh order elements.
+9
+Conclusions and outlook
+The intrinsic behaviour of the relaxed micromorphic model is revealed by the analytical solutions to boundary
+value problems. Clearly, the continuum exhibits hyperbolic and trigonometric solutions, which are not easily
+approximated by low order finite elements. The example provided in Section 8.2 demonstrates that cubic and
+higher order finite elements yield excellent results in approximate solutions of the model.
+The polytopal template methodology introduced in [50] allows to easily and flexibly construct H (curl)-
+conforming vectorial finite elements that inherit many of the characteristics of an underlying H 1-conforming
+basis, which can be chosen independently. In this work, we made use of Bernstein-B´ezier polynomials. The
+latter boast optimal complexity properties manifesting in the form of sum factorization. The natural decom-
+position of their multi-variate versions into multiplications of univariate Bernstein base functions via the Duffy
+transformation allows to construct optimal iterators for their evaluation using recursion formulas. Further, this
+characteristic makes the use of dual numbers in the computation of their derivatives ideal. Finally, the intrinsic
+order of traversal induced by the factorization is exploited optimally by the choice of clock-wise orientation
+of the reference element. The consequence of these combined features is a high-performance hp-finite element
+program.
+32
+
+(a)
+(b)
+(c)
+Figure 8.5: Displacement field of the Cauchy model on the coarsest mesh of 48 finite elements of the tenth order
+(a) and depictions of the meshes with 384 (b) and 3072 (c) elements, respectively.
+The ability of the relaxed micromorphic model to interpolate between the energies of homogeneous materials
+and materials with an underlying micro-structure using the characteristic length scale parameter Lc is demon-
+strated in Section 8.3. It is also shown that in order to correctly capture the span of energies for the values of
+Lc either fine-discretizations or higher order elements are required.
+The excellent performance of the proposed higher order finite elements in the linear static case is a precur-
+sor for their application in the dynamic setting, which is important since the relaxed micromorphic model is
+often employed in the computation of elastic waves (e.g., for acoustic metamaterials), where solutions for high
+frequency ranges are commonly needed.
+The proposed computational scheme is lacking in its description of curved geometries. Due to the consistent
+coupling condition, this can easily lead to errors emanating from the boundary. Consequently, a topic for future
+works would be the investigation of curved finite elements [20,21] and their behaviour with respect to the model.
+Acknowledgements
+Angela Madeo and Gianluca Rizzi acknowledge support from the European Commission through the funding
+of the ERC Consolidator Grant META-LEGO, N◦ 101001759.00
+Patrizio Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational
+Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project-
+No. 440935806.
+10
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+36
+

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+Deep Learning for bias-correcting comprehensive
+high-resolution Earth system models
+Philipp Hess1,2, Stefan Lange2, and Niklas Boers1,2,3
+1Earth System Modelling, School of Engineering & Design, Technical University of Munich,
+Munich, Germany
+2Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
+3Global Systems Institute and Department of Mathematics, University of Exeter, Exeter, UK
+Key Points:
+• A generative adversarial network is shown to improve daily precipitation fields from
+a state-of-the-art Earth system model.
+• Biases in long-term temporal distributions are strongly reduced by the generative
+adversarial network.
+• Our network-based approach can be complemented with quantile mapping to fur-
+ther improve precipitation fields.
+–1–
+arXiv:2301.01253v1  [physics.ao-ph]  16 Dec 2022
+
+Abstract
+The accurate representation of precipitation in Earth system models (ESMs) is crucial for
+reliable projections of the ecological and socioeconomic impacts in response to anthropogenic
+global warming. The complex cross-scale interactions of processes that produces precipi-
+tation are challenging to model, however, inducing potentially strong biases in ESM fields,
+especially regarding extremes. State-of-the-art bias correction methods only address errors
+in the simulated frequency distributions locally, at every individual grid cell. Improving
+unrealistic spatial patterns of the ESM output, which would require spatial context, has
+not been possible so far. Here, we show that a post-processing method based on physically
+constrained generative adversarial networks (GANs) can correct biases of a state-of-the-art,
+CMIP6-class ESM both in local frequency distributions and in the spatial patterns at once.
+While our method improves local frequency distributions equally well as gold-standard bias-
+adjustment frameworks it strongly outperforms any existing methods in the correction of
+spatial patterns, especially in terms of the characteristic spatial intermittency of precipita-
+tion extremes.
+1 Introduction
+Precipitation is a crucial climate variable and changing amounts, frequencies, or spatial
+distributions have potentially severe ecological and socioeconomic impacts.
+With global
+warming projected to continue in the coming decades, assessing the impacts of changes
+in precipitation characteristics is an urgent challenge (Wilcox & Donner, 2007; Boyle &
+Klein, 2010; IPCC, 2021). Climate impact models are designed to assess the impacts of
+global warming on, for example, ecosystems, crop yields, vegetation and other land-surface
+characteristics, infrastructure, water resources, or the economy in general (Kotz et al., 2022),
+using the output of climate or Earth system models (ESMs) as input. Especially for reliable
+assessments of the ecological and socioeconomic impacts, accurate ESM precipitation fields
+to feed the impact models are therefore crucial.
+ESMs are integrated on spatial grids with finite resolution. The resolution is limited
+by the computational resources that are necessary to perform simulations on decadal to
+centennial time scales. Current state-of-the-art ESMs have a horizontal resolution on the
+order of 100km, in exceptional cases going down to 50km. Smaller-scale physical processes
+that are relevant for the generation of precipitation operate on scales below the size of
+individual grid cells. These can therefore not be resolved explicitly in ESMs and have to
+included as parameterizations of the resolved prognostic variables. These include droplet
+interactions, turbulence, and phase transitions in clouds that play a central role in the
+generation of precipitation.
+The limited grid resolution hence introduces errors in the simulated precipitation fields,
+leading to biases in short-term spatial patterns and long-term summary statistics. These
+biases need to be addressed prior to passing the ESM precipitation fields to impact mod-
+els. In particular, climate impact models are often developed and calibrated with input
+data from reanalysis data rather than ESM simulations. These reanalyses are created with
+data assimilation routines and combine various observations with high-resolution weather
+models. They hence provide a much more realistic input than the ESM simulations and
+statistical bias correction methods are necessary to remove biases in the ESM simulations
+output and to make them more similar to the reanalysis data for which the impact models
+are calibrated. Quantile mapping (QM) is a standard technique to correct systematic errors
+in ESM simulations. QM estimates a mapping between distributions from historical sim-
+ulations and observations that can thereafter be applied to future simulations in order to
+provide more accurate simulated precipitation fields to impact models (D´equ´e, 2007; Tong
+et al., 2021; Gudmundsson et al., 2012; Cannon et al., 2015).
+State-of-the-art bias correction methods such as QM are, however, confined to address
+errors in the simulated frequency distributions locally, i.e., at every grid cell individually.
+–2–
+
+Unrealistic spatial patterns of the ESM output, which would require spatial context, have
+therefore so far not been addressed by postprocessing methods. For precipitation this is
+particularly important because it has characteristic high intermittency not only in time,
+but also in its spatial patterns. Mulitvariate bias correction approaches have recently been
+developed, aiming to improve spatial dependencies (Vrac, 2018; Cannon, 2018). However,
+these approaches are typically only employed in regional studies, as the dimension of the
+input becomes too large for global high-resolution ESM simulations. Moreover, such meth-
+ods have been reported to suffer from instabilities and overfitting, while differences in their
+applicability and assumptions make them challenging to use (Fran¸cois et al., 2020).
+Here, we employ a recently introduced postprocessing method (Hess et al., 2022) based
+on a cycle-consistent adversarial network (CycleGAN) to consistently improve both local
+frequency distributions and spatial patterns of state-of-art high-resolution ESM precipita-
+tion fields. Artificial neural networks from computer vision and image processing have been
+successfully applied to various tasks in Earth system science, ranging from weather forecast-
+ing (Weyn et al., 2020; Rasp & Thuerey, 2021) to post-processing (Gr¨onquist et al., 2021;
+Price & Rasp, 2022), by extracting spatial features with convolutional layers (LeCun et al.,
+2015). Generative adversarial networks (Goodfellow et al., 2014) in particular have emerged
+as a promising architecture that produces sharp images that are necessary to capture the
+high-frequency variability of precipitation (Ravuri et al., 2021; Price & Rasp, 2022; Harris et
+al., 2022). GANs have been specifically developed to be trained on unpaired image datasets
+(Zhu et al., 2017). This makes them a natural choice for post-processing the output of cli-
+mate projections, which – unlike weather forecasts – are not nudged to follow the trajectory
+of observations; due to the chaotic nature of the atmosphere small deviations in the initial
+conditions or parameters lead to exponentially diverging trajectories (Lorenz, 1996). As a
+result, numerical weather forecasts lose their deterministic forecast skill after approximately
+two weeks at most and century-scale climate simulations do not agree with observed daily
+weather records. Indeed the task of climate models is rather to produce accurate long-term
+statistics that to agree with observations.
+We apply our CycleGAN approach to correct global high-resolution precipitation simu-
+lations of the GFDL-ESM4 model (Krasting et al., 2018) as a representative ESM from the
+Climate Model Intercomparison Project phase 6 (CMIP6). So far, GANs-based approaches
+have only been applied to postprocess ESM simulations either in a regional context (Fran¸cois
+et al., 2021), or to a very-low-resolution global ESM (Hess et al., 2022). We show here that
+a suitably designed CycleGAN is capable of improving even the distributions and spatial
+patterns of precipitation fields from a state-of-the-art comprehensive ESM, namely GFDL-
+ESM4. In particular, in contrast to rather specific existing methods for postprocessing ESM
+output for climate impact modelling, we will show that the CycleGAN is general and can
+readily be applied to different ESMs and observational datasets used as ground truth.
+In order to assure that physical conservation laws are not violated by the GAN-based
+postprocessing, we include a suitable physical constraint, enforcing that the overall global
+sum of daily precipitation values is not changed by the GAN-based transformations; es-
+sentially, this assures that precipitation is only spatially redistributed (see Methods). By
+framing bias correction as an image-to-image translation task, our approach corrects both
+spatial patterns of daily precipitation fields on short time scales and temporal distributions
+aggregated over decadal time scales. We evaluate the skill to improve spatial patterns and
+temporal distributions against the gold-standard ISIMIP3BASD framework (Lange, 2019),
+which relies strongly on QM.
+Quantifying the “realisticness” of spatial precipitation patterns is a key problem in
+current research (Ravuri et al., 2021). We use spatial spectral densities and the fractal
+dimension of spatial patterns as a measure to quantify the similarity of intermittent and un-
+paired precipitation fields. We will show that our CycleGAN is indeed spatial context-aware
+and strongly improves the characteristic intermittency in spatial precipitation patterns. We
+–3–
+
+will also show that our CycleGAN combined with a subseqeunt application of ISIMIP3BASD
+routine leads to the best overall performance.
+2 Results
+We evaluate our CycleGAN method on two different tasks and time scales. First, the
+correction of daily rainfall frequency distributions at each grid cell locally, aggregated from
+decade-long time series. Second, we quantify the ability to improve spatial patterns on daily
+time scales. Our GAN approach is compared to the raw GFDL-ESM4 model output, as well
+as to the ISIMIP3BASD methodology applied to the GFDL-ESM4 output.
+2.1 Temporal distributions
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Histogram
+a
+0
+98.4
+99.7
+99.94
+99.98
+99.993
+99.997
+W5E5v2 precipitation percentiles
+W5E5v2
+GFDL-ESM4
+ISIMIP3BASD
+GAN
+GAN (unconstrained)
+GAN-ISIMIP3BASD
+0
+25
+50
+75
+100
+125
+150
+Precipitation [mm/d]
+10
+8
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+Absolute error
+b
+Figure
+1: Histograms of relative precipitation frequencies over the entire globe and test
+period (2004-2014). (a) The histograms are shown for the W5E5v2 ground truth (black),
+GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), unconstrained GAN (orange),
+and the constrained-GAN-ISIMIP3BASD combination (blue).
+(b) Distances of the his-
+tograms to the W5E5v2 ground truth are shown for the same models as in (a). Percentiles
+corresponding to the W5E5v2 precipitation values are given on the second x-axis at the
+top. Note that GFDL-ESM4 overestimates the frequencies of strong and extreme rainfall
+events. All compared methods show similar performance in correcting the local frequency
+distributions.
+–4–
+
+We compute global histograms of relative precipitation frequencies using daily time
+series (Fig. 1a). The GFDL-ESM4 model overestimates frequencies in the tail, namely for
+events above 50 mm/day (i.e., the 99.7th percentile). Our GAN-based method as well as
+ISIMIP3BASD and the GAN-ISIMIP3BASD combination correct the histogram to match
+the W5E5v2 ground truth equally well, as can be also seen in the absolute error of the
+histograms (Fig. 1b).
+Comparing the differences in long-term averages of precipitation per grid cell (Fig. 2
+and Methods), large biases are apparent in the GFDL-ESM4 model output, especially in
+the tropics. The double-peaked Intertropical Convergence Zone (ITCZ) bias is visible. The
+double-ITCZ bias can also be inferred from the latitudinal profile of the precipitation mean
+in Fig. 3.
+Table 1 summarizes the annual biases shown in Fig. 2 as absolute averages, and addi-
+tionally for the four seasons. The GAN alone reduces the annual bias of the GFDL-ESM4
+model by 38.7%. The unconstrained GAN performs better than the physically constrained
+one, with bias reductions of 50.5%. As expected, the ISIMIP3BASD gives even better results
+for correcting the local mean, since it is specifically designed to accurately transform the
+local frequency distributions. It is therefore remarkable that applying the ISIMIP3BASD
+procedure on the constrained GAN output improves the post-processing further, leading to
+a local bias reduction of the mean by 63.6%, compared to ISIMIP3BASD with 59.4%. For
+seasonal time series the order in which the methods perform is the same as for the annual
+data.
+Besides the error in the mean, we also compute differences in the 95th percentile for each
+grid cell, shown in Fig. S1 and as mean absolute errors in Table 1. Also in this case of heavy
+precipitation values we find that ISIMIP3BASD outperforms the GAN, but that combining
+GAN and ISIMIP3BASD leads to best agreement of the locally computed quantiles.
+Table 1: The globally averaged absolute value of the grid cell-wise difference in the long-
+term precipitation average, as well as the 95th percentile, between the W5E5v2 ground truth
+and GFDL-ESM4, ISIMIP3BASD, GAN, unconstrained GAN, and the GAN-ISIMIP3BASD
+combination for annual and seasonal time series (in [mm/day]). The relative improvement
+over the raw GFDL-ESM4 climate model output is shown as percentages for each method.
+Season
+Percentile
+GFDL-
+ESM4
+ISIMIP3-
+BASD
+%
+GAN
+%
+GAN
+(unconst.)
+%
+GAN-
+ISIMIP3-
+BASD
+%
+Annual
+-
+0.535
+0.217
+59.4
+0.328
+38.7
+0.265
+50.5
+0.195
+63.6
+DJF
+-
+0.634
+0.321
+49.4
+0.395
+37.7
+0.371
+41.5
+0.308
+51.4
+MAM
+-
+0.722
+0.314
+56.5
+0.419
+42.0
+0.378
+47.6
+0.285
+60.5
+JJA
+-
+0.743
+0.289
+61.1
+0.451
+39.3
+0.357
+52.0
+0.280
+62.3
+SON
+-
+0.643
+0.327
+49.1
+0.409
+36.4
+0.362
+43.7
+0.306
+52.4
+Annual
+95th
+2.264
+1.073
+52.6
+1.415
+37.5
+1.213
+46.4
+0.945
+58.3
+DJF
+95th
+2.782
+1.496
+46.2
+1.725
+38.0
+1.655
+40.5
+1.432
+48.5
+MAM
+95th
+2.948
+1.482
+49.7
+1.805
+38.8
+1.661
+43.7
+1.337
+54.6
+JJA
+95th
+2.944
+1.366
+53.6
+1.852
+37.1
+1.532
+48.0
+1.247
+57.6
+SON
+95th
+2.689
+1.495
+44.4
+1.741
+35.3
+1.592
+40.8
+1.366
+49.2
+–5–
+
+Figure
+2: Bias in the long-term average precipitation over the entire test set between
+the W5E5v2 ground truth (a) and GFDL-ESM4 (b), ISIMIP3BASD (c), GAN (d), uncon-
+strained GAN (e) and the GAN-ISIMIP3BASD combination (f).
+2.2 Spatial patterns
+We compare the ability of the GAN to improve spatial patterns based on the W5E5v2
+ground truth, against the GFDL-ESM4 simulations and the ISIMIP3BASD method applied
+to the GFDL-ESM4 simulations. To model realistic precipitation fields, the characteristic
+spatial intermittency needs to be captured accurately.
+We compute the spatial power spectral density (PSD) of global precipitation fields,
+averaged over the test set for each method. GFDL-ESM4 shows noticeable deviations from
+W5E5v2 in the PSD (Fig. 4). Our GAN can correct these over the entire range of wave-
+–6–
+
+W5E5v2 mean [mm/d]
+GFDL-ESM4
+a
+b
+N.09
+0°
+S.09
+0
+ISIMIP3BASD
+GAN
+N.09
+0°
+60°S
+GAN (unconstrained)
+GAN-ISIMIP3BASD
+e
+f
+N.09
+0°
+S.09
+120°W
+60°W
+0
+60°E
+120°E
+120°W
+60°W
+0°
+60°E
+120°E
+7.5
+-7.5 -5.0 -2.5
+0.0
+2.5
+5.0
+Bias [mm/d]80 S
+60 S
+40 S
+20 S
+0
+20 N
+40 N
+60 N
+80 N
+Latitude
+0
+1
+2
+3
+4
+5
+6
+7
+Mean precipitation [mm/d]
+W5E5v2
+GFDL-ESM4: MAE = 0.241
+ISIMIP3BASD: MAE = 0.120
+GAN: MAE = 0.226
+GAN (unconstrained): MAE = 0.102
+GAN-ISIMIP3BASD: MAE = 0.068
+Figure
+3: Precipitation averaged over longitudes and the entire test set period from the
+W5E5v2 ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN
+(cyan), unconstrained GAN (orange) and the GAN-ISIMIP3BASD combination (blue). To
+quantify the differences between the shown lines, we show their mean absolute error w.r.t
+the W5E5v2 ground truth in the legend. These values are different from the ones shown in
+Table 1 as the average is taken here over the longitudes without their absolute value. The
+GAN-ISIMIP3BASD approach shows the lowest error.
+lengths, closely matching the W5E5v2 ground truth. Improvements over ISIMIP3BASD
+are especially pronounced in the range of high frequencies (low wavelengths), which are
+responsible for the intermittent spatial variability of daily precipitation fields. Adding the
+physical constraint to the GAN does not affect the ability to produce realistic PSD distribu-
+tions. After applying ISIMIP3BASD to the GAN-processed fields, most of the improvements
+generated by the GAN are retained, as shown by the GAN-ISIMIP3BASD results.
+For a second way to quantifying how realistic the simulated and post-processed pre-
+cipitation fields are, with a focus on high-frequency spatial intermittency, we investigate
+the fractal dimension (Edgar & Edgar, 2008) of the lines separating grid cells with daily
+rainfall sums above and below a given quantile threshold (see Methods). For a sample and
+qualitative comparison of precipitation fields over the South American continent see Fig. S2.
+The daily spatial precipitation fields are first converted to binary images using a quantile
+threshold. The respective quantiles are determined from the precipitation distribution over
+the entire test set period and globe. The mean of the fractal dimension computed with box-
+counting (see Methods) (Lovejoy et al., 1987; Meisel et al., 1992; Husain et al., 2021) for each
+time slice is then investigated (Fig. 5). Both the GFDL-ESM4 simulations themselves and
+the results of applying the ISIMIP3BASD post-processing to them exhibit spatial patterns
+with a lower fractal dimension than the W5E5v2 ground truth, implying too low spatial
+intermittency. In contrast, the GAN translates spatial fields simulated by GFDL-ESM4 in
+a way that results in closely matching fractal dimensions over the entire range of quantiles.
+3 Discussion
+Postprocessing climate projections is a fundamentally different task from postprocessing
+weather forecast simulations (Hess et al., 2022). In the latter case, data-driven postprocess-
+ing methods, e.g. based on deep learning, to minimize differences between paired samples
+–7–
+
+128
+256
+512
+1024
+2048
+4096
+8192
+Wavelength [km]
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+PSD [a.u]
+W5E5v2
+GFDL-ESM4
+ISIMIP3BASD
+GAN
+GAN (unconstrained)
+GAN-ISIMIP3BASD
+Figure 4: The power spectral density (PSD) of the spatial precipitation fields is shown as
+an average over all samples in the test set for the W5E5v2 ground truth (black) and GFDL-
+ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan, dashed), unconstrained GAN (orange,
+dashed-dotted) and the constrained-GAN-ISIMIP3BASD combination (blue, dotted). The
+GANs and W5E5v2 ground truth agree so closely that they are indistinguishable. In contrast
+to ISIMIP3BASD, the GAN can correct the intermittent spectrum accurately over the entire
+range down to the smallest wavelengths.
+of variables such as spatial precipitation fields (Hess & Boers, 2022). Beyond time scales of
+a few days, however, the chaotic nature of the atmosphere leads to exponentially diverging
+trajectories, and for climate or Earth system model output there is no observation-based
+ground truth to directly compare to. We therefore frame the post-processing of ESM projec-
+tions, with applications for subsequent 195 impact modelling in mind, as an image-to-image
+translation task with unpaired samples.
+To this end we apply a recently developed postprocessing method based on physically
+constrained CycleGANs to global simulations of a state-of-the-art, high-resolution ESM
+from the CMIP6 model ensemble, namely the GFDL-ESM4 (Krasting et al., 2018; O' Neill
+et al., 2016). We evaluate our method against the gold-standard bias correction framework
+ISIMIP3BASD. Our model can be trained on unpaired samples that are characteristic for
+climate simulations. It is able to correct the ESM simulations in two regards: temporal
+distributions over long time scales, including extremes in the distrivutions’ tails, as well
+as spatial patterns of individual global snap shots of the model output. The latter is not
+possible with established methods.
+Our GAN-based approach is designed as a general
+framework that can be readily applied to different ESMs and observational target datasets.
+This is in contrast to existing bias-adjustment methods that are often tailored to specific
+applications.
+We chose to correct precipitation because it is arguably one of the hardest variables
+to represent accurately in ESMs. So far, GANs have only been applied to regional studies
+or low-resolution global ESMs (Fran¸cois et al., 2021; Hess et al., 2022). The GFDL-ESM4
+model simulations are hence chosen in order to test if our CycleGAN approach would lead
+–8–
+
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Quantile
+1.3
+1.4
+1.5
+1.6
+1.7
+Fractal dimension
+W5E5v2
+GFDL-ESM4: MAE = 0.048
+ISIMIP3BASD: MAE = 0.037
+GAN: MAE = 0.002
+GAN (unconstrained): MAE = 0.002
+GAN-ISIMIP3BASD: MAE = 0.004
+Figure 5: The fractal dimension (see Methods) of binary global precipitation fields is com-
+pared as averages for different quantile thresholds.
+Results are shown for the W5E5v2
+ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), un-
+constrained GAN (orange, dashed), and the GAN-ISIMIP3BASD combination (blue). The
+GAN can accurately reproduce the fractal dimension of the W5E5v2 ground truth spatial
+precipitation fields over all quantile thresholds, clearly outperforming the ISIMIP3BASD
+basline.
+to improvements even when postprocessing global high-resolution simulations of one of the
+most complex and sophisticated ESMs to date. In the same spirit, we evaluate our ap-
+proach against a very strong baseline given by the state-of-the-art bias correction framework
+ISIMIP3BASD, which is based on a trend-preserving QM method (Lange, 2019).
+Comparing long-term summary statistics, our method yields histograms of relative pre-
+cipitation frequencies that very closely agree with corresponding histograms from reanalysis
+data (Fig. 1). The means that the extremes in the far end of the tail are accurately cap-
+tured, with similar skill to the ISIMIP3BASD baseline that is mainly designed for this task.
+Differences in the grid cell-wise long-term average show that the GAN skillfully reduces bi-
+ases (Fig. 2); in particular, the often reported double-peaked ITCZ bias of the GFDL-ESM4
+simulations, which is a common feature of most climate models (Tian & Dong, 2020), is
+strongly reduced (Fig. 3). The ISIMIP3BASD method - being specifically designed for this
+- produces slightly lower biases for grid-cell-wise averages than the GAN; we show that
+combining both methods by first applying the GAN and then the ISIMIP3BASD procedure
+leads to the overall best performance.
+Regarding the correction of spatial patterns of the modelled precipitation fields, we
+compare the spectral density and fractal dimensions of the spatial precipitation fields. Our
+results show that indeed only the GAN can capture the characteristic spatial intermittency
+of precipitation closely (Figs. 4 and 5). We believe that the measure of fractal dimension
+is also relevant for other fields such as nowcasting and medium-range weather forecasting,
+where blurriness in deep learning-based predictions is often reported (Ravuri et al., 2021)
+and needs to be further quantified.
+–9–
+
+Post-processing methods for climate projections have to be able to preserve the trends
+that result from the non-stationary dynamics of the Earth system on long-time scales. We
+have therefore introduced the architecture constraint of preserving the global precipitation
+amount on every day in the climate model output (Hess et al., 2022). We find that this does
+not affect the quality of the spatial patterns that are produced by our CycleGAN method.
+However, the skill of correcting mean error biases is slightly reduced by the constraint. This
+can be expected in part as the constraint is constructed to follow the global mean of the
+ESM. Hence, biases in the global ESM mean can influence the constrained GAN. This also
+motivates our choice to demonstrate the combination of the constrained GAN with the QM-
+based ISIMIP3BASD procedure, since it can be applied to future climate scenarios, making
+it more suitable for actual applications than the unconstrained architecture.
+There are several directions to further develop or approach. The architecture employed
+here has been built for equally spaced two-dimensional images. Extending the CycleGAN
+architecture to perform convolutions on the spherical surface, e.g. using graph neural net-
+works, might lead to more efficient and accurate models. Moreover, GANs are comparably
+difficult to train, which could make it challenging to identify suitable network architectures.
+Using large ensembles of climate simulations could provide additional training data that
+could further improve the performance. Another straightforward extension of our method
+would be the inclusion of further input variables or the prediction additional high-impact
+physical variables, such as near-surface temperatures that are also important for regional
+impact models.
+4 Methods
+4.1 Training data
+We use global fields of daily precipitation with a horizontal resolution of 1◦ from the
+GFDL-ESM4 Earth system model (Krasting et al., 2018) and the W5E5v2 reanalysis prod-
+uct (Cucchi et al., 2020; WFDE5 over land merged with ERA5 over the ocean (W5E5 v2.0),
+2021) as observation-based ground truth.
+The W5E5v2 dataset is based on the ERA5
+(Hersbach et al., 2020) reanalysis and has been bias-adjusted using the Global Precipitation
+Climatology Centre (GPCC) full data monthly product v2020 (Schneider et al., 2011) over
+land and the Global Precipitation Climatology Project (GPCP) v2.3 dataset (Huffman et
+al., 1997) over the ocean. Both datasets have been regridded to the same 1◦ horizontal
+resolution using bilinear interpolation following (Beck et al., 2019). We split the dataset
+into three periods for training (1950-2000), validation (2001-2003), and testing (2004-2014).
+This corresponds to 8030 samples for training, 1095 for validation, and 4015 for testing.
+During pre-processing, the training data is log-transformed with ˜x = log(x+ϵ)−log(ϵ) with
+ϵ = 0.0001, following Rasp and Thuerey (2021), to account for zeros in the transform. The
+data is then normalized to the interval [−1, 1] following (Zhu et al., 2017).
+4.2 Cycle-consistent generative adversarial networks
+This section gives a brief overview of the CycleGAN used in this study. We refer to
+(Zhu et al., 2017; Hess et al., 2022) for a more comprehensive description and discussion.
+Generative adversarial networks learn to generate images that are nearly indistinguishable
+from real-world examples through a two-player game (Goodfellow et al., 2014).
+In this
+set-up, a first network G, the so-called generator, produces images with the objective to
+fool a second network D, the discriminator, which has to classify whether a given sample
+is generated (“fake”) or drawn from a real-world dataset (“real”). Mathematically this can
+be formalized as
+G∗ = min
+G
+max
+D
+LGAN(D, G),
+(1)
+–10–
+
+with G∗ being the optimal generator network. The loss function LGAN(D, G) can be defined
+as
+LGAN(D, G) = Ey∼py(y)[log(D(y))] + Ex∼px(x)[log(1 − D(G(x)))],
+(2)
+where py(y) is the distribution of the real-world target data and samples from px(x) are
+used as inputs by G to produce realistic images. The CycleGAN (Zhu et al., 2017) consists
+of two generator-discriminator pairs, where the generators G and F learn inverse mappings
+between two domains X and Y . This allows to define an additional cycle-consistency loss
+that constraints the training of the networks, i.e.
+Lcycle(G, F) = Ex∼px(x)[||F(G(x)) − x||1]
+(3)
++ Ey∼py(y)[||G(F(y)) − y||1].
+It measures the error caused by a translation cycle of an image to the other domain and
+back. Further, an additional loss term is introduced to regularize the networks to be close
+to an identity mapping with,
+Lident(G, F) = Ex∼px(x)[||G(x) − x||1]
+(4)
++ Ey∼py(y)[||F(y) − y||1].
+In practice, the log-likelihood loss can be replaced by a mean squared error loss to facilitate
+a more stable training.
+Further, the generator loss is reformulated to be minimized by
+inverting the labels, i.e.
+LGenerator = Ex∼px(x)[(DX(G(x)) − 1)2]
++ Ey∼py(y)[(DY (F(y)) − 1)2]
+(5)
++ λLcycle(G, F) + ˜λLident(G, F),
+where λ and ˜λ are set to 10 and 5 respectively following (Zhu et al., 2017). The corresponding
+loss term for the discriminator networks is given by
+LDiscriminator = Ey∼py(y)[(DY (y) − 1)2] + Ex∼px(x)[(DX(G(x)))2]
+(6)
++ Ex∼px(x)[(DX(x) − 1)2] + Ey∼py(y)[(DY (F(y)))2].
+(7)
+The weights of the generator and discriminator networks are then optimized with the ADAM
+(Kingma & Ba, 2014) optimizer using a learning rate of 2e−4 and updated in an alternating
+fashion. We train the network for 350 epochs and a batch size of 1, saving model checkpoints
+every other epoch. We evaluate the checkpoints on the validation dataset to determine the
+best model instance.
+4.3 Network Architectures
+Both the generator and discriminator have fully convolutional architectures. The gen-
+erator uses ReLU activation functions, instance normalization, and reflection padding. The
+discriminator uses leaky ReLU activations with slope 0.2 instead, together with instance
+normalization. For a more detailed description, we refer to our previous study (Hess et al.,
+2022). The network architectures in this study are the same, only with a change in the
+number of residual layers in the generator network from 6 to 7.
+The final layer of the generator can be constrained to preserve the global sum of the
+input, i.e. by rescaling
+˜yi = yi
+�Ngrid
+i
+xi
+�Ngrid
+i
+yi
+,
+(8)
+–11–
+
+where xi and yi are grid cell values of the generator input and output respectively and
+Ngrid is the number of grid cells. The generator without this constraint will be referred
+to as unconstrained in this study. The global physical constraint enforces that the global
+daily precipitation sum is not affected by the CycleGAN postprocessing and hence remains
+identical to the original value from the GFDL-ESM4 simualtions. This is motivated by the
+observation that large-scale average trends in precipitation follow the Clausius-Clapeyron
+relation (Traxl et al., 2021), which is based on thermodynamic relations and hence can be
+expected to be modelled well in GFDL-ESM4.
+4.4 Quantile mapping-based bias adjustment
+We compare the performance of our GAN-based method to the bias adjustment method
+ISMIP3BASD v3.0.1 (Lange, 2019, 2022) that has been developed for phase 3 of the Inter-
+Sectoral Impact Model Intercomparison Project (Warszawski et al., 2014; Frieler et al.,
+2017). This state-of-the-art bias-adjustment method is based on a trend-preserving quantile
+mapping (QM) framework. It represents a very strong baseline for comparison as it has
+been developed prior to this study and used not only in ISIMIP3 but also to prepare many
+of the climate projections that went into the Interactive Atlas produced as part of the 6th
+assessment report of working group 1 of the Intergovernmental Panel on Climate Change
+(IPCC, https://interactive-atlas.ipcc.ch/). In QM, a transformation between the cumulative
+distribution functions (CDFs) of the historical simulation and observations is fitted and then
+applied to future simulations. The CDFs can either be empirical or parametric, the latter
+being a Bernoulli-gamma distribution for the precipitation in this study. The CFDs are
+fitted and mapped for each grid cell and day of the year separately. For bias-adjusting the
+GFDL-ESM4 simulation, parametric QM was found to give the best results, while empirical
+CDFs are used in combination with the GAN.
+To evaluate the methods in this study we define the grid cell-wise bias as the difference
+in long-term averages as,
+Bias(ˆy, y) = 1
+T
+T
+�
+t=1
+ˆyt − 1
+T
+T
+�
+t=1
+yt,
+(9)
+where T is the number of time steps, ˆyt and ˆyt the modelled and observed precipitation
+respectively at time step t.
+4.5 Evaluating spatial patterns
+Quantifying how realistic spatial precipitation fields are is an ongoing research question
+in itself, which has become more important with the application of deep learning to weather
+forecasting and post-processing. In these applications, neural networks often achieve error
+statistics and skill scores competitive with physical models, while the output fields can
+at the same time show unphysical characteristics, such as blurring or excessive smoothing.
+Ravuri et al. (2021) compare the spatial intermittency, which is characteristic of precipitation
+fields, using the power spectral density (PSD) computed from the spatial fields; in the latter
+study, the PSD-based quantification was complemented by interviews with a large number
+of meteorological experts. We propose the fractal dimension of binary precipitation fields
+as an alternative to quantify how realistic the patterns are.
+We compute the fractal dimension via the box-counting algorithm (Lovejoy et al., 1987;
+Meisel et al., 1992). It quantifies how spatial patterns, for example coastlines (Husain et
+al., 2021), change with the scale of measurement. The box-counting algorithm divides the
+image into squares and counts the number of squares that cover the binary pattern of
+interest, Nsquares. The size of the squares, i.e. the scale of measurement, is then reduced
+iteratively by a factor s. The fractal dimension Dfractal can then be determined from the
+slope of the resulting log-log scaling, i.e.,
+–12–
+
+Dfractal = log(Nsquares)
+log(s)
+.
+(10)
+Competing interests
+The authors declare no competing interests.
+Data availability
+The W5E5 data is available for download at https://doi.org/10.48364/ISIMIP.342217.
+The GFDL-ESM4 data can be downloaded at https://esgf-node.llnl.gov/projects/
+cmip6/.
+Code availability
+The Python code for processing and analysing the data, together with the PyTorch
+Lightning (Falcon et al., 2019) code is available at https://github.com/p-hss/earth
+system model gan bias correction.git. The ISIMIP3BASD code in (Lange, 2022) is
+used for this study.
+Acknowledgments
+NB and PH acknowledge funding by the Volkswagen Foundation, as well as the European
+Regional Development Fund (ERDF), the German Federal Ministry of Education and Re-
+search and the Land Brandenburg for supporting this project by providing resources on the
+high performance computer system at the Potsdam Institute for Climate Impact Research.
+N.B. acknowledges funding by the European Union’s Horizon 2020 research and innovation
+programme under grant agreement No 820970 and under the Marie Sklodowska-Curie grant
+agreement No. 956170, as well as from the Federal Ministry of Education and Research
+under grant No. 01LS2001A. SL acknowledges funding from the European Union’s Horizon
+2022 research and innovation programme under grant agreement no. 101081193 Optimal
+High Resolution Earth System Models for Exploring Future Climate Changes (OptimESM).
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+conference on computer vision (pp. 2223–2232).
+–15–
+
+Supporting Information for ”Deep Learning for
+bias-correcting comprehensive high-resolution Earth
+system models”
+Philipp Hess1,2, Stefan Lange2, and Niklas Boers1,2,3
+1Earth System Modelling, School of Engineering & Design, Technical University of Munich, Munich, Germany
+2Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
+3Global Systems Institute and Department of Mathematics, University of Exeter, Exeter, UK
+Contents of this file
+1. Figure S1 to S2
+January 4, 2023, 1:28am
+arXiv:2301.01253v1  [physics.ao-ph]  16 Dec 2022
+
+X - 2
+:
+Figure S1.
+Bias maps as in Fig.
+2 but with the 95th percentile instead of the mean.
+Global mean absolute errors (MAEs) are given in the respective titles. Combining the GAN with
+ISIMIP3BASD achieves the lowest error compared to the other methods.
+January 4, 2023, 1:28am
+
+W5E5v2 95th percentile [mm/d]
+GFDL-ESM4: MAE = 2.264
+b
+a
+N.09
+0°
+S.09
+0
+25
+50
+120°W
+60°W
+0°
+60°E
+120°E
+120°W
+60°W
+0°
+60°E
+120°E
+ISIMIP3BASD: MAE = 1.073
+GAN: MAE = 
+1.415
+d
+N.09
+0°
+S.09
+120°W
+60°W
+.0
+60°E
+120°E
+60°W
+0°
+60°E
+120°W
+120°E
+GAN (unconstrained): MAE
+1.213
+GAN-ISIMIP3BASD: MAE
+0.945
+=
+e
+2
+120°W
+60°W
+0°
+60°E
+120°E
+120°W
+60°W
+0°
+60°E
+120°E
+-20 -i5 -i0 -5　0
+5
+10
+15
+20
+Differences in the 95th percentile [mm/d]:
+X - 3
+a
+50°S
+25°S
+0°
+100°W
+75°W
+50°W
+25°W
+W5E5v2
+c
+50°S
+25°S
+0°
+100°W
+75°W
+50°W
+25°W
+ISIMIP3BASD
+b
+50°S
+25°S
+0°
+100°W
+75°W
+50°W
+25°W
+GFDL-ESM4
+d
+50°S
+25°S
+0°
+100°W
+75°W
+50°W
+25°W
+GAN-ISIMIP3BASD
+5
+10
+15
+20
+25
+30
+35
+Precipitation [mm/d]
+Figure S2.
+Qualitative comparison of precipitation fields at the same date (December 21st
+2014) over the South American continent. The region is used for a comparison of the fractal
+dimension in binary precipitation patterns.
+January 4, 2023, 1:28am
+

ENAyT4oBgHgl3EQfSPf9/content/tmp_files/2301.00085v1.pdf.txt

@@ -0,0 +1,1124 @@
+arXiv:2301.00085v1  [math.CO]  31 Dec 2022
+On the chromatic number of random regular
+hypergraphs
+Patrick Bennett∗
+Department of Mathematics,
+Western Michigan University
+Kalamazoo MI 49008
+Alan Frieze†
+Department of Mathematical Sciences,
+Carnegie Mellon University,
+Pittsburgh PA 15213.
+Abstract
+We estimate the likely values of the chromatic and independence numbers of the
+random r-uniform d-regular hypergraph on n vertices for fixed r, large fixed d, and
+n → ∞.
+1
+Introduction
+The study of the chromatic number of random graphs has a long history. It begins with the
+work of Bollob´as and Erd˝os [6] and Grimmett and McDiarmid [13] who determined χ(Gn,p),
+p constant to within a factor 2, w.h.p. Matula [17] reduced this to a factor of 3/2. Then
+we have the discovery of martingale concentration inequalities by Shamir and Spencer [18]
+leading to the breakthrough by Bollob´as [5] who determined χ(Gn,p) asymptotically for p
+constant.
+The case of p → 0 proved a little more tricky, but �Luczak [15] using ideas from Frieze [10]
+and [17] determined χ(Gn,p), p = c/n asymptotically for large c. �Luczak [16] showed that
+w.h.p. χ(Gn,p), p = c/n took one of two values. It was then that the surprising power of
+the second moment method was unleashed by Achlioptas and Naor [3]. Since then there has
+been much work tightening our estimates for the k-colorability threshold, k ≥ 3 constant.
+See for example Coja-Oghlan [7].
+Random regular graphs of low degree were studied algorithmically by several authors e.g.
+Achlioptas and Molloy [2] and by Shi and Wormald [19]. Frieze and �Luczak [12] introduced
+∗Research supported in part by Simons Foundation Grant #426894.
+†Research supported in part by NSF Grant DMS1661063
+1
+
+a way of using our knowledge of χ(Gn,p), p = c/n to tackle χ(Gn,r) where Gn,r denotes a
+random r-regular graph and where p = r/n. Subsequently Achlioptas and Moore [2] showed
+via the second moment method that w.h.p. χ(Gn,r) was one of 3 values. This was tightened
+basically to one value by Coja-Oghlan, Efthymiou and Hetterich [8].
+For random hypergraphs, Krivelevich and Sudakov [14] established the asymptotic chromatic
+number for χ(Hr(n, p) for
+�n−1
+r−1
+�
+p sufficiently large. Here Hr(n, p) is the binomial r-uniform
+hypergraph where each of the
+�n
+r
+�
+possible edges is included with probability p. There are
+several possibilities of a proper coloring of the vertices of a hypergraph. Here we concentrate
+on the case where a vertex coloring is proper if no edge contains vertices of all the same color.
+Dyer, Frieze and Greehill [9] and Ayre, Coja-Oghlan and Greehill [1] established showed that
+w.h.p. χ(Hr(n, p) took one or two values. When it comes to what ew denote by χ(Hr(n, d),
+a random d-regular, r-uniform hypergraph, we are not aware of any results at all. In this
+paper we extend the approach of [12] to this case:
+Theorem 1. For all fixed r and ε > 0 there exists d0 = d0(r, ε) such that for any fixed
+d ≥ d0 we have that w.h.p.
+�������
+χ(Hr(n, d)) −
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�������
+≤ ε,
+�������
+α(Hr(n, d)) −
+�
+r log d
+(r−1)d
+�
+1
+r−1 n
+�
+r log d
+(r−1)d
+�
+1
+r−1 n
+�������
+≤ ε
+(1)
+Here α refers to the independence number of a hypergraph.
+2
+Preliminaries
+2.1
+Tools
+We will be using the following forms of Chernoff’s bound (see, e.g., [11]).
+Lemma 2 (Chernoff bound). Let X ∼ Bin(n, p). Then for all 0 < λ < np
+P(|X − np| ≥ λ) ≤ 2 exp
+�
+− λ2
+3np
+�
+.
+(2)
+Lemma 3 (McDiarmid’s inequality). Let X = f(⃗Z) where ⃗Z = (Z1, . . . Zt) and the Zi are
+independent random variables. Assume the function f has the property that whenever ⃗z, ⃗w
+differ in only one coordinate we have |f(⃗z) − f(⃗w)| ≤ c. Then for all λ > 0 we have
+P(|X − E[X]| ≥ λ) ≤ 2 exp
+�
+− λ2
+2c2t
+�
+.
+(3)
+Bal and the first author [4] showed the following.
+2
+
+Theorem 4 (Claim 4.2 in [4]). Fix r ≥ 3, d ≥ 2, and 0 < c < r−1
+r . Let z2 be the unique
+positive number such that
+z2
+�
+(z2 + 1)r−1 − zr−1
+2
+�
+(z2 + 1)r − zr
+2
+= c
+(4)
+and let
+z1 =
+d
+r [(z2 + 1)r − zr
+2].
+(5)
+Let h(x) = x log x. If it is the case that
+h
+�d
+r
+�
++ h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d
+r log z1 − dc log z2 < 0
+(6)
+then w.h.p. α(Hr(n, d)) < cn.
+Krivelevich and Sudakov [14] proved the following.
+Theorem 5 (Theorem 5.1 in [14]). For all fixed r and ε > 0 there exists d0 = d0(r, ε) such
+that whenever D = D(p) :=
+�n−1
+r−1
+�
+p ≥ d0 we have that
+�������
+χ(Hr(n, p)) −
+�
+(r−1)D
+r log D
+�
+1
+r−1
+�
+(r−1)D
+r log D
+�
+1
+r−1
+�������
+≤ ε,
+�������
+α(Hr(n, p)) −
+�
+r log D
+(r−1)D
+�
+1
+r−1 n
+�
+r log D
+(r−1)D
+�
+1
+r−1 n
+�������
+≤ ε
+with probability at least 1 ��� o(1/n).
+3
+Proof
+In this section we prove Theorem 1. First we give an overview. We show in Subsection 3.1
+that the upper bound on α follows from Theorem 4 and some straightforward calculations.
+Then the lower bound on χ follows as well. Thus we will be done once we prove the upper
+bound on χ (since that proves the lower bound on α). This will be in Subsection 3.2. For
+that we follow the methods of Frieze and �Luczak [12].
+We will assume r ≥ 3 since Frieze and �Luczak [12] covered the graph case. We will use
+standard asymptotic notation, and we will use big-O notation to suppress any constants
+depending on r but not d. Thus, for example we will write r = O(1) and d−1 = O(1) but
+not d = O(1). This is convenient for us because even though our theorem is for fixed d, it
+requires d to be sufficiently large.
+3
+
+3.1
+Upper bound on the independence number
+We will apply Theorem 4 to show an upper bound on α(Hr(n, d)). Fix ε, r (but not d) and
+let c = c(d) := (1 + ε)
+�
+r log d
+(r−1)d
+�
+1
+r−1. Let z2 be as defined in (4) and z1 be as defined in (5).
+We see that
+Lemma 6.
+z2 =
+c
+1 − c + O (cr)
+Proof. After some algebra, we re-write (4) as
+z2 −
+zr
+2
+(1 + z2)r−1 =
+c
+1 − c.
+and the claim follows.
+Now we check (6).
+h
+�d
+r
+�
++ h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d
+r log z1 − dc log z2
+=d
+r log
+�d
+r
+�
++ dc log(dc) + d(1 − c) log(d(1 − c)) − c log c − (1 − c) log(1 − c)
+− d log d − d
+r log z1 − dc log z2
+=dc log
+�
+c
+(1 − c)z2
+�
++ d
+r log [(z2 + 1)r − zr
+2] + d log(1 − c) − c log c − (1 − c) log(1 − c). (7)
+Now note that the first term of (7) is
+dc log
+�
+c
+(1 − c)z2
+�
+= dc log
+�
+c
+(1 − c)
+�
+c
+1−c + O (cr+1)
+�
+�
+= dc log
+�
+1
+1 + O (cr)
+�
+= O
+�
+dcr+1�
+.
+The second term of (7) is
+d
+r log [(z2 + 1)r − zr
+2] = d
+r log
+��
+1
+1 − c + O
+�
+cr+1��r
+−
+�
+c
+1 − c + O
+�
+cr+1��r�
+= d
+r log
+��
+1
+1 − c
+�r �
+1 − cr + O
+�
+cr+1���
+= d
+r log
+�
+1
+1 − c
+�r
++ d
+r log
+�
+1 − cr + O
+�
+cr+1��
+= −d log(1 − c) − d
+r cr + O
+�
+dcr+1�
+.
+4
+
+The last term of (7) is
+(1 − c) log(1 − c) = O(c).
+Therefore (7) becomes
+− d
+r cr − c log c + O
+�
+c + dcr+1�
+= − c
+�d
+rcr−1 + log c
+�
++ O
+�
+c + dcr+1�
+= − c
+�
+d
+r (1 + ε)r−1 r log d
+(r − 1)d + log
+�
+(1 + ε)
+� r log d
+(r − 1)d
+�
+1
+r−1��
++ O
+�
+c + dcr+1�
+= − c
+�
+(1 + ε)r−1 log d
+r − 1 − log d
+r − 1 + O(log log d)
+�
++ O
+�
+c + dcr+1�
+= − Ω (c log d) .
+It follows from Theorem 4 that w.h.p.
+α(Hr(n, d)) ≤ (1 + ε)
+� r log d
+(r − 1)d
+�
+1
+r−1
+.
+(8)
+3.2
+Upper bound on the chromatic number
+Our proof of the upper bound uses the method of Frieze and �Luczak [12]. We will generate
+Hr(n, d) in a somewhat complicated way. The way we generate it will allow us to use known
+results on Hr(n, p) due to Krivelevich and Sudakov [14].
+Set
+m :=
+�d − d1/2 log d
+r
+�
+n.
+(9)
+Let H∗
+r(n, m) be an r-uniform multi-hypergraph with m edges, where each multi-edge consists
+of r independent uniformly random vertices chosen with replacement.
+We will generate
+H∗
+r(n, m) as follows. We have n sets (“buckets” ) V1, . . . Vn and a set of rm points P :=
+{p1, . . . prm}. We put each point pi into a uniform random bucket Vφ(i) independently. We
+let R = {R1, . . . , Rm} be a uniform random partition of P into sets of size r. Of course, the
+idea here is that the buckets Vi represent vertices and the parts of the partition R represent
+edges. Thus Ri defines a hyper-edge {φ(j) : j ∈ Ri} for i = 1, 2, . . . , m. We denote the
+hypergraph defined by R by HR.
+Note that since r ≥ 3 the expected number of pairs of multi-edges in H∗
+r(n, m) is at most
+�n
+r
+��m
+2
+� �
+1
+�n
+r
+�
+�2
+= O
+�m2
+nr
+�
+= O(n−1).
+5
+
+Thus, w.h.p. there are no multi-edges. Now the expected number of “loops” (edges containing
+the same vertex twice) is at most
+nm
+�r
+2
+� �1
+n
+�2
+= O(1).
+Thus w.h.p. there are at most log n loops. We now remove all multi-edges and loops, and
+say that M is the (random) number of edges remaining, where m − log n ≤ M ≤ m. The
+remaining hypergraph is distributed as H(n, M), the random hypergraph with M edges
+chosen uniformly at random without replacement. Next we estimate the chromatic number
+of Hr(n, M).
+Claim 1. W.h.p. we have
+�������
+χ(Hr(n, M)) −
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�������
+≤ ε
+2,
+�������
+α(Hr(n, M)) −
+�
+r log D
+(r−1)D
+�
+1
+r−1 n
+�
+r log D
+(r−1)D
+�
+1
+r−1 n
+�������
+≤ ε
+2.
+Proof. We will use Theorem 5 together with a standard argument for comparing Hr(n, p)
+with Hr(n, m). Set p := m/
+�n
+r
+�
+and apply Theorem 5 with ε replaced with ε/4 so we get
+�������
+χ(Hr(n, p)) −
+�
+(r−1)D
+r log D
+�
+1
+r−1
+�
+(r−1)D
+r log D
+�
+1
+r−1
+�������
+≤ ε
+4
+(10)
+with probability at least 1 − o(1/n). Note that here
+D =
+�n − 1
+r − 1
+�
+p =
+�n − 1
+r − 1
+�
+m/
+�n
+r
+�
+= rm/n = d − d1/2 log d.
+Now since d, D can be chosen to be arbitrarily large and d = D + O(D1/2 log D) we can
+replace D with d in (10) without changing the left hand side by more than ε/4 to obtain
+�������
+χ(Hr(n, p)) −
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�������
+≤ ε
+2
+(11)
+with probability at least 1−o(1/n). But now note that with probability Ω(n−1/2) the number
+of edges in Hr(n, p) is precisely M. Thus we have that
+�������
+χ(Hr(n, M)) −
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�
+(r−1)d
+r log d
+�
+1
+r−1
+�������
+≤ ε
+2
+with probability at least 1 − o(n−1/2). This proves the first inequality, and the second one
+follows similarly.
+6
+
+Now we will start to transform Hr(n, m) to the random regular hypergraph Hr(n, d). This
+transformation will involve first removing some edges from vertices of degree larger than d,
+and then adding some edges to vertices of degree less than d. We define the rank of a point
+pi ∈ Vj, to be the number of points pi′ ∈ Vj such that i′ ≤ i. We form a new set of points
+P ′ ⊆ P and a partition R′ of P ′ as follows. For any Rk ∈ R containing a point with rank
+more than d, we delete Rk from R and delete all points of Rk from P. Note that each bucket
+contains at most r points of P ′. Note also that R′ is a uniform random partition of P ′. We
+let HR′ be the natural hypergraph associated with R′.
+Now we would like to put some more points into the buckets until each bucket has exactly d
+points, arriving at some set of points P ′′ ⊇ P ′. We would also like a uniform partition R′′ of
+P ′′ into sets of size r, and we would like R′′ to have many of the same parts as R′. We will
+accomplish this by constructing a sequence P ′
+1 := P ′ ⊆ P ′
+2 ⊆ . . . ⊆ P ′
+ℓ =: P ′′ of point sets
+and a sequence R′
+1 := R′, R′
+2, . . . , R′
+ℓ =: R′′ where R′
+j is a uniform random partition of P ′
+j.
+We construct P ′
+j+1, R′
+j+1 from P ′
+j, R′
+j as follows. Suppose |R′
+j| = a (in other words R′
+j has
+a parts), so |P ′
+j| = ra. P ′
+j+1 will simply be P ′
+j plus r new points. Now we will choose a
+random value K ∈ {1, . . . , r} using the distribution P[K = k] = qk(a), where qk(a) is defined
+as follows.
+Definition 1. Consider a random partition of ra+r points into a+1 parts of size r, and fix
+some set Q of r points. Then for 1 ≤ k ≤ r, the number qk(a) is defined to be the probability
+that Q meets exactly k parts of the partition.
+We will then remove a uniform random set of K − 1 parts from R′
+j, leaving Kr points in
+P ′
+j+1 which are not in any remaining part of R′
+j. We partition those points into K parts of
+size r such that each part contains at least one new point (each such partition being equally
+likely), arriving at our partition R′
+j+1.
+We claim that R′
+j+1 is a uniform random partition of P ′
+j+1 into parts of size r. Indeed, first
+consider the r new points that are in P ′
+j+1 which were not in P ′
+j. The probability that a
+uniform random partition of P ′
+j+1 would have exactly k parts containing at least one new
+point is qk. So we can generate such a random partition as follows: first choose a random
+value K with P[K = k] = qk; next we choose a uniform random set of (K − 1)r points from
+P ′
+j; next we choose a partition of the set of points consisting of P ′
+j+1 \ P ′
+j together with the
+points from P ′
+j we chose in the last step, where the partition we choose is uniformly random
+from among all partitions such that each part contains at least one point of P ′
+j+1 \P ′
+j; finally,
+we choose a uniform partition of the rest of the points. In our case this partition of the rest
+of the points comprises the current partition of the “unused” (a − K + 1)r points. At the
+end of this process we have that HR′′ is distributed as Hr(n, d).
+7
+
+3.2.1
+Bounding the number of low degree vertices in HR′
+We define some sets of buckets. We show that w.h.p. there are few small buckets i.e few
+vertices of low degree in the hypregraph HR′. Let S0 be the buckets with at most d−3d1/2 log d
+points of P ′, and let S1 be the buckets with at most d−2d1/2 log d points of P. Let S2 be the
+set of buckets that, when we remove points from P ′ to get P, have at least d1/2 log d points
+removed. Then S0 ⊆ S1 ∪ S2. Our goal is to bound the probability that S0 is too large.
+Fix a bucket Vj and let X ∼ Bin
+�
+rm, 1
+n
+�
+be the number of points of P in Vj. Then the
+probability that Vj is in S1 satisfies
+P[Vj ∈ S1] = P
+�
+X ≤ d − 2d1/2 log d
+�
+= P
+�
+X − rm
+n ≤ −d1/2 log d
+�
+≤ exp
+�
+−
+d log2 d
+3(d − d1/2 log d)
+�
+= exp
+�
+−Ω
+�
+log2 d
+��
+,
+where for our inequality we have used the Chernoff bound (Lemma 2). Therefore E[|S0|] ≤
+exp
+�
+−Ω
+�
+log2 d
+��
+n. Now we argue that |S1| is concentrated using McDiarmid’s inequality
+(Lemma 3). For our application we let X = |S1| which is a function (say f) of the vector
+(Z1, . . . Zrm) where Zi tells us which bucket the ith point of P went into. Moving a point
+from one bucket to another can only change |S1| by at most 1 so we use c = 1. Thus we get
+the bound
+P(|X − E[X]| ≥ n2/3) ≤ 2 exp
+�
+− n4/3
+2rm
+�
+= o(1).
+(12)
+Now we handle S2. For 1 ≤ j ≤ n let Yj be the number of parts Rk ∈ R such that Rk
+contains a point in the bucket Vj as well as a point in some bucket Vj′ where |Vj′| > d. Note
+that if Vj ∈ S2 then Yj ≥ d1/2 log d. We view Rk as a set of r points, say {q1, . . . , qr} each
+going into a uniform random bucket. Say qi goes to bucket Vji. The probability that Rk is
+counted by Yj is at most
+rP[j1 = j and |Vj1| > d] + r(r − 1)P[j1 = j and |Vj2| > d]
+= r
+nP[|Vj1| > d
+��j1 = j] + r(r − 1)
+n
+P[|Vj2| > d
+��j1 = j]
+≤ r2
+n P[|Vj1| > d
+��j1 = j]
+≤ r2
+n P[Bin(rm − 1, 1/n) ≥ d] = r2
+n exp
+�
+−Ω
+�
+log2 d
+��
+.
+Thus we have
+E[Yj] = m · r2
+n exp
+�
+−Ω
+�
+log2 d
+��
+≤ rd exp
+�
+−Ω
+�
+log2 d
+��
+= rd1/2 exp
+�
+−Ω
+�
+log2 d
+��
+and so Markov’s inequality gives us
+P
+�
+Yj ≥ d1/2 log d
+�
+≤ rd exp
+�
+−Ω
+�
+log2 d
+��
+d1/2 log d
+= exp
+�
+−Ω
+�
+log2 d
+��
+8
+
+and so E[|S2|] = n exp
+�
+−Ω
+�
+log2 d
+��
+. We use McDiarmid’s inequality once more, this time
+with X = |S2|.
+A change in choice of bucket changes |S2| by at most one and so (12)
+continues to hold. Thus
+|S0| = n exp
+�
+−Ω
+�
+log2 d
+��
+.
+w.h.p.
+3.2.2
+A property of independent subsets of Hr(n, m)
+Fix 1 ≤ j ≤ r − 1. Set
+a :=
+�
+1 + ε
+2
+� � r log d
+(r − 1)d
+�
+1
+r−1
+,
+κj := 10d
+r
+�r
+j
+�
+aj,
+p := d(r − 1)!
+nr−1
+.
+The expected number of independent sets A in Hr(n, p) of size at most an such that there
+are κjn edges each having j vertices in A is at most
+an
+�
+s=1
+�n
+s
+�
+(1 − p)(s
+r)
+��s
+j
+�� n
+r−j
+�
+κjn
+�
+pκjn
+≤
+an
+�
+s=1
+exp
+
+
+s log
+�en
+s
+�
+−
+�s
+r
+�
+p + κjn log
+
+e(an)j
+j!
+nr−j
+(r−j)!p
+κjn
+
+
+
+
+
+=
+an
+�
+s=1
+exp
+�
+s log
+�en
+s
+�
+−
+�s
+r
+�
+p + κjn log
+�eaj
+10
+��
+≤ an · exp
+��
+log
+�e
+a
+�
+− 10d
+r
+�r
+j
+�
+aj−1 log
+�10
+e
+��
+an
+�
+= o(1/n)
+where the last line follows since as d → ∞ we have
+log
+�e
+a
+�
+∼
+1
+r − 1 log d
+and
+10d
+r
+�r
+j
+�
+aj−1 log
+�10
+e
+�
+= Ω
+�
+d
+r−j
+j−1 log− j−1
+r−1 d
+�
+≫ log d.
+Thus with probability 1 − o(1/n), Hr(n, p) has a coloring using (1 + ε/2)
+�
+(r−1)d
+r log d
+�
+1
+r−1 colors
+such that for each color class A and for each 1 ≤ j ≤ r − 1 there are at most κjn edges with
+j vertices in A. The hypergraph Hr(n, m), m =
+�n
+r
+�
+p will have this property w.h.p..
+3.2.3
+Transforming HR′ into Hr(n, d)
+Now we will complete the transformation to the random regular hypergraph Hr(n, d). We
+are open to the possibility that doing so will render our coloring no longer proper, since this
+9
+
+process will involve changing some edges which might then be contained in a color class. We
+will keep track of how many such “bad” edges there are and then repair our coloring at the
+end.
+We have to add at most (3d1/2 log d + d exp
+�
+−Ω
+�
+log2 d
+��
+)n < (4d1/2 log d)n points, which
+takes at most as many steps.
+For each color class A of HR′ define XA,j = XA,j(i) to
+be the number of edges with j vertices in A at step i. We have already established that
+XA,j(0) ≤ κjn. This follows from Section 3.2.2 and the fact that we have removed edges
+from H(n, m) to obtain HR′. Let Ei be the event that at step i we have that for each color
+class A and for each 1 ≤ j ≤ r − 1 we have XA,j(i) ≤ 2κjn. Then, assuming Ei holds, the
+probability that XA,j increases at step i is at most
+�
+1≤k≤r, jℓ≥1
+j1+···+jk=j
+�
+1≤ℓ≤k
+2κjℓn
+nd/r =
+�
+1≤k≤r, jℓ≥1
+j1+···+jk=j
+�
+1≤ℓ≤k
+20
+� r
+jk
+�
+ajk ≤
+�
+1≤k≤r, jℓ≥1
+j1+···+jk=j
+20r2r2aj ≤ 40r2r2aj.
+Also, the largest possible increase in XA,j in one step is r. Thus, the final value of XA,j
+after at most (4d1/2 log d)n steps is stochastically dominated by κjn + rY where Y
+∼
+Bin
+�
+(4d1/2 log d)n, 40r2r2aj�
+. An easy application of the Chernoff bound tells us
+P (Y > 2E[Y ]) ≤ exp(−Ω(n)).
+(13)
+Note that here
+2E[Y ]
+κjn
+= 8d1/2 log d · 40r2r2ajn
+10d
+�r
+j
+�
+ajn/r
+= O(d−1/2 log d) < 1
+for sufficiently large d. Thus, using (13) and the union bound over all color classes A, we
+have w.h.p. the final value of XA,j is at most κjn + 2E[Y ] ≤ 2κjn for all 1 ≤ j ≤ r − 1.
+Now we address “bad” edges, i.e. edges contained in a color class. Assuming Ei holds, the ex-
+pected number of new edges contained in any color class at step i is at most r(40)r2r2+2rar =
+O
+�� log d
+d
+�
+r
+r−1�
+(because it would have to be one of the colors of one of the vertices we are
+adding points to). Thus the expected number of bad edges created in (4d1/2 log d)n steps
+is stochastically dominated by Z ∼ r · Bin
+�
+(4d1/2 log d)n, O
+��log d
+d
+�
+r
+r−1� �
+.
+Another easy
+application of Chernoff shows that w.h.p. Z ≤ 2E[Z] = O(d−1/2n).
+We repair the coloring as follows. First we uncolor one vertex from each bad edge, and let
+the set of uncolored vertices be U where |U| = u = O
+�
+d−1/2n
+�
+. Let
+δ := ε
+2
+�(r − 1)d
+r log d
+�
+1
+r−1
+.
+We claim that for every S ⊆ U, |S| = s, the hypergraph induced on S has at most δs/r
+edges. This will complete our proof since it implies that the minimum degree is at most δ
+and so U can be recolored using a fresh set of δ colors, yielding a coloring of Hr(n, d) using
+10
+
+at most
+χ(Hr(n, M)) + δ ≤
+�
+1 + ε
+2
+� �(r − 1)d
+r log d
+�
+1
+r−1
++ ε
+2
+�(r − 1)d
+r log d
+�
+1
+r−1
+= (1 + ε)
+�(r − 1)d
+r log d
+�
+1
+r−1
+colors. The expected number of sets S with more than δs/r edges is at most
+�
+1≤s≤u
+�n
+s
+���ds
+r
+�
+δs/r
+�
+1
+�dn
+r
+��dn−r
+r
+�
+. . .
+�dn−δs+r
+r
+�
+≤
+�
+1≤s≤u
+�ne
+s
+�s �(dse/r)re
+δs/r
+�δs/r
+(r!)δs/r
+(dn − δs)δs
+≤
+�
+1≤s≤u
+�
+ne
+s
+�
+dse
+(dn − δs)r
+�δ �er · r!
+δs
+�δ/r�s
+.
+(14)
+Now for 1 ≤ s ≤ √n the term in (14) is at most
+�
+O(n) ·
+�
+O(n−1/2)
+�δ · O(1)
+�s
+= o(1/n)
+since δ can be made arbitrarily large by choosing d large. Meanwhile for √n ≤ s ≤ u we
+have that the term in (14) is at most
+�
+O(n1/2) · O(1) ·
+�
+O(n−1/2)
+�δ/r�s
+= o(1/n).
+Now since (14) has O(n) terms the whole sum is o(1) and we are done. This completes the
+proof of Theorem 1.
+4
+Summary
+We have asymptotically computed the chromatic number of random r-uniform, d-regular
+hypergraphs when proper colorings mean that no edge is mono-chromatic. It would seem
+likely that the approach we took would extend to other definitions of proper coloring. We
+have not attempted to use second moment calculations to further narrow our estimates.
+These would seem to be two natural lines of further research.
+References
+[1] P. Ayre, A. Coja-Oghlan and C. Greenhill, Hypergraph coloring up to condensation,
+Random Structures and Algorithms 54 (2019) 615 - 652.
+11
+
+[2] D. Achlioptas and C. Moore, The Chromatic Number of Random Regular Graphs,
+In Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds) Approximation, Random-
+ization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM AP-
+PROX 2004 2004. Lecture Notes in Computer Science, vol 3122. Springer, Berlin, Hei-
+delberg. Approximation, Randomization, and Combinatorial Optimization. Algorithms
+and Techniques (2004) 219–228.
+[3] D. Achlioptas and A. Naor, The two possible values of the chromatic number of a
+random graph, Annals of Mathematics 162 (2005) 1335-1351.
+[4] D. Bal and P. Bennett, The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs.
+[5] B. Bollob´as, The chromatic number of random graphs, Combinatorica 8 (1988) 49-55.
+[6] B. Bollob´as and P. Erd˝os, Cliques in random graphs, Mathematical Proceedings of the
+Cambridge Philosophical Society 80 (1976) 419-427.
+[7] A. Coja-Oghlan, Upper-Bounding the k-Colorability Threshold by Counting Covers,
+Electronic Journal of Combinatorics 20 (2013).
+[8] A. Coja-Oghlan, C. Efthymiou and S. Hetterich, On the chromatic number of random
+regular graphs, Journal of Combinatorial Theory B 116 (2016) 367-439.
+[9] M. Dyer, A.M. Frieze and C. Greenhill, On the chromatic number of a random hyper-
+graph, Journal of Combinatorial Theorey B 113 (2015) 68-122.
+[10] A.M. Frieze, On the independence number of random graphs, Discrete Mathematics 81
+(1990) 171-176.
+[11] A.M. Frieze and M. Karo´nski, Introduction to Random Graphs, Cambridge University
+Press, 2015.
+[12] A.M. Frieze and T. �Luczak, On the independence and chromatic numbers of random
+regular graphs, Journal of Combinatorial Theory. Series B 54 (1992) 123-132.
+[13] G. Grimmett and C. McDiarmid, On colouring random graphs, Mathematical Proceed-
+ings of the Cambridge Philosophical Society 77 (1975) 313-324.
+[14] M. Krivelevich and B. Sudakov, The chromatic numbers of random hypergraphs, Ran-
+dom Structures Algorithms 12 (1998) 381-403.
+[15] T. �Luczak, The chromatic number of random graphs, Combinatorica 11 (19990) 45-54.
+[16] T. �Luczak, A note on the sharp concentration of the chromatic number of random
+graphs, Combinatorica 11 (1991) 295-297.
+[17] D. Matula, Expose-and-Merge Exploration and the Chromatic Number of a Random
+Graph, Combinatorica 7 (1987) 275-284.
+12
+
+[18] E. Shamir and J. Spencer, Sharp concentration of the chromatic number od random
+graphs Gn,p, Combinatorica 7 (1987) 121-129.
+[19] L. Shi and N. Wormald, Coloring random regular graphs, Combinatorics, Probability
+and Computing 16 (2007) 459-494.
+13
+

ENAyT4oBgHgl3EQfSPf9/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf,len=361
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='00085v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='CO]  31 Dec 2022  On the chromatic number of random regular  hypergraphs  Patrick Bennett∗  Department of Mathematics,  Western Michigan University  Kalamazoo MI 49008  Alan Frieze†  Department of Mathematical Sciences,  Carnegie Mellon University,  Pittsburgh PA 15213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Abstract  We estimate the likely values of the chromatic and independence numbers of the  random r-uniform d-regular hypergraph on n vertices for fixed r, large fixed d, and  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  1  Introduction  The study of the chromatic number of random graphs has a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' It begins with the  work of Bollob´as and Erd˝os [6] and Grimmett and McDiarmid [13] who determined χ(Gn,p),  p constant to within a factor 2, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Matula [17] reduced this to a factor of 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Then  we have the discovery of martingale concentration inequalities by Shamir and Spencer [18]  leading to the breakthrough by Bollob´as [5] who determined χ(Gn,p) asymptotically for p  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  The case of p → 0 proved a little more tricky, but �Luczak [15] using ideas from Frieze [10]  and [17] determined χ(Gn,p), p = c/n asymptotically for large c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' �Luczak [16] showed that  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' χ(Gn,p), p = c/n took one of two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' It was then that the surprising power of  the second moment method was unleashed by Achlioptas and Naor [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Since then there has  been much work tightening our estimates for the k-colorability threshold, k ≥ 3 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  See for example Coja-Oghlan [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Random regular graphs of low degree were studied algorithmically by several authors e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Achlioptas and Molloy [2] and by Shi and Wormald [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Frieze and �Luczak [12] introduced  ∗Research supported in part by Simons Foundation Grant #426894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  †Research supported in part by NSF Grant DMS1661063  1  a way of using our knowledge of χ(Gn,p), p = c/n to tackle χ(Gn,r) where Gn,r denotes a  random r-regular graph and where p = r/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Subsequently Achlioptas and Moore [2] showed  via the second moment method that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' χ(Gn,r) was one of 3 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This was tightened  basically to one value by Coja-Oghlan, Efthymiou and Hetterich [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  For random hypergraphs, Krivelevich and Sudakov [14] established the asymptotic chromatic  number for χ(Hr(n, p) for  �n−1  r−1  �  p sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Here Hr(n, p) is the binomial r-uniform  hypergraph where each of the  �n  r  �  possible edges is included with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' There are  several possibilities of a proper coloring of the vertices of a hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Here we concentrate  on the case where a vertex coloring is proper if no edge contains vertices of all the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Dyer, Frieze and Greehill [9] and Ayre, Coja-Oghlan and Greehill [1] established showed that  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' χ(Hr(n, p) took one or two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' When it comes to what ew denote by χ(Hr(n, d),  a random d-regular, r-uniform hypergraph, we are not aware of any results at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' In this  paper we extend the approach of [12] to this case:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' For all fixed r and ε > 0 there exists d0 = d0(r, ε) such that for any fixed  d ≥ d0 we have that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  �������  χ(Hr(n, d)) −  �  (r−1)d  r log d  �  1  r−1  �  (r−1)d  r log d  �  1  r−1  �������  ≤ ε,  �������  α(Hr(n, d)) −  �  r log d  (r−1)d  �  1  r−1 n  �  r log d  (r−1)d  �  1  r−1 n  �������  ≤ ε  (1)  Here α refers to the independence number of a hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  2  Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='1  Tools  We will be using the following forms of Chernoff’s bound (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=', [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Lemma 2 (Chernoff bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let X ∼ Bin(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Then for all 0 < λ < np  P(|X − np| ≥ λ) ≤ 2 exp  �  − λ2  3np  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (2)  Lemma 3 (McDiarmid’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let X = f(⃗Z) where ⃗Z = (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Zt) and the Zi are  independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Assume the function f has the property that whenever ⃗z, ⃗w  differ in only one coordinate we have |f(⃗z) − f(⃗w)| ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Then for all λ > 0 we have  P(|X − E[X]| ≥ λ) ≤ 2 exp  �  − λ2  2c2t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (3)  Bal and the first author [4] showed the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  2  Theorem 4 (Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Fix r ≥ 3, d ≥ 2, and 0 < c < r−1  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let z2 be the unique  positive number such that  z2  �  (z2 + 1)r−1 − zr−1  2  �  (z2 + 1)r − zr  2  = c  (4)  and let  z1 =  d  r [(z2 + 1)r − zr  2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (5)  Let h(x) = x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' If it is the case that  h  �d  r  �  + h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d  r log z1 − dc log z2 < 0  (6)  then w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' α(Hr(n, d)) < cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Krivelevich and Sudakov [14] proved the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Theorem 5 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='1 in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' For all fixed r and ε > 0 there exists d0 = d0(r, ε) such  that whenever D = D(p) :=  �n−1  r−1  �  p ≥ d0 we have that  �������  χ(Hr(n, p)) −  �  (r−1)D  r log D  �  1  r−1  �  (r−1)D  r log D  �  1  r−1  �������  ≤ ε,  �������  α(Hr(n, p)) −  �  r log D  (r−1)D  �  1  r−1 n  �  r log D  (r−1)D  �  1  r−1 n  �������  ≤ ε  with probability at least 1 − o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  3  Proof  In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' First we give an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We show in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='1  that the upper bound on α follows from Theorem 4 and some straightforward calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Then the lower bound on χ follows as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus we will be done once we prove the upper  bound on χ (since that proves the lower bound on α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This will be in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' For  that we follow the methods of Frieze and �Luczak [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We will assume r ≥ 3 since Frieze and �Luczak [12] covered the graph case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We will use  standard asymptotic notation, and we will use big-O notation to suppress any constants  depending on r but not d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus, for example we will write r = O(1) and d−1 = O(1) but  not d = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This is convenient for us because even though our theorem is for fixed d, it  requires d to be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='1  Upper bound on the independence number  We will apply Theorem 4 to show an upper bound on α(Hr(n, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Fix ε, r (but not d) and  let c = c(d) := (1 + ε)  �  r log d  (r−1)d  �  1  r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let z2 be as defined in (4) and z1 be as defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We see that  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  z2 =  c  1 − c + O (cr)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' After some algebra, we re-write (4) as  z2 −  zr  2  (1 + z2)r−1 =  c  1 − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Now we check (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  h  �d  r  �  + h(dc) + h(d(1 − c)) − h(c) − h(1 − c) − h(d) − d  r log z1 − dc log z2  =d  r log  �d  r  �  + dc log(dc) + d(1 − c) log(d(1 − c)) − c log c − (1 − c) log(1 − c)  − d log d − d  r log z1 − dc log z2  =dc log  �  c  (1 − c)z2  �  + d  r log [(z2 + 1)r − zr  2] + d log(1 − c) − c log c − (1 − c) log(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' (7)  Now note that the first term of (7) is  dc log  �  c  (1 − c)z2  �  = dc log  �  c  (1 − c)  �  c  1−c + O (cr+1)  �  �  = dc log  �  1  1 + O (cr)  �  = O  �  dcr+1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  The second term of (7) is  d  r log [(z2 + 1)r − zr  2] = d  r log  ��  1  1 − c + O  �  cr+1��r  −  �  c  1 − c + O  �  cr+1��r�  = d  r log  ��  1  1 − c  �r �  1 − cr + O  �  cr+1���  = d  r log  �  1  1 − c  �r  + d  r log  �  1 − cr + O  �  cr+1��  = −d log(1 − c) − d  r cr + O  �  dcr+1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  4  The last term of (7) is  (1 − c) log(1 − c) = O(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Therefore (7) becomes  − d  r cr − c log c + O  �  c + dcr+1��  = − c  �d  rcr−1 + log c  �  + O  �  c + dcr+1�  = − c  �  d  r (1 + ε)r−1 r log d  (r − 1)d + log  �  (1 + ε)  � r log d  (r − 1)d  �  1  r−1��  + O  �  c + dcr+1�  = − c  �  (1 + ε)r−1 log d  r − 1 − log d  r − 1 + O(log log d)  �  + O  �  c + dcr+1�  = − Ω (c log d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  It follows from Theorem 4 that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  α(Hr(n, d)) ≤ (1 + ε)  � r log d  (r − 1)d  �  1  r−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (8)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2  Upper bound on the chromatic number  Our proof of the upper bound uses the method of Frieze and �Luczak [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We will generate  Hr(n, d) in a somewhat complicated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' The way we generate it will allow us to use known  results on Hr(n, p) due to Krivelevich and Sudakov [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Set  m :=  �d − d1/2 log d  r  �  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (9)  Let H∗  r(n, m) be an r-uniform multi-hypergraph with m edges, where each multi-edge consists  of r independent uniformly random vertices chosen with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We will generate  H∗  r(n, m) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We have n sets (“buckets” ) V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Vn and a set of rm points P :=  {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' prm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We put each point pi into a uniform random bucket Vφ(i) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We  let R = {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' , Rm} be a uniform random partition of P into sets of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Of course, the  idea here is that the buckets Vi represent vertices and the parts of the partition R represent  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus Ri defines a hyper-edge {φ(j) : j ∈ Ri} for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We denote the  hypergraph defined by R by HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Note that since r ≥ 3 the expected number of pairs of multi-edges in H∗  r(n, m) is at most  �n  r  ��m  2  � �  1  �n  r  �  �2  = O  �m2  nr  �  = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  5  Thus, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' there are no multi-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Now the expected number of “loops” (edges containing  the same vertex twice) is at most  nm  �r  2  � �1  n  �2  = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Thus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' there are at most log n loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We now remove all multi-edges and loops, and  say that M is the (random) number of edges remaining, where m − log n ≤ M ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' The  remaining hypergraph is distributed as H(n, M), the random hypergraph with M edges  chosen uniformly at random without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Next we estimate the chromatic number  of Hr(n, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' we have  �������  χ(Hr(n, M)) −  �  (r−1)d  r log d  �  1  r−1  �  (r−1)d  r log d  �  1  r−1  �������  ≤ ε  2,  �������  α(Hr(n, M)) −  �  r log D  (r−1)D  �  1  r−1 n  �  r log D  (r−1)D  �  1  r−1 n  �������  ≤ ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We will use Theorem 5 together with a standard argument for comparing Hr(n, p)  with Hr(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Set p := m/  �n  r  �  and apply Theorem 5 with ε replaced with ε/4 so we get  �������  χ(Hr(n, p)) −  �  (r−1)D  r log D  �  1  r−1  �  (r−1)D  r log D  �  1  r−1  �������  ≤ ε  4  (10)  with probability at least 1 − o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Note that here  D =  �n − 1  r − 1  �  p =  �n − 1  r − 1  �  m/  �n  r  �  = rm/n = d − d1/2 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Now since d, D can be chosen to be arbitrarily large and d = D + O(D1/2 log D) we can  replace D with d in (10) without changing the left hand side by more than ε/4 to obtain  �������  χ(Hr(n, p)) −  �  (r−1)d  r log d  �  1  r−1  �  (r−1)d  r log d  �  1  r−1  �������  ≤ ε  2  (11)  with probability at least 1−o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' But now note that with probability Ω(n−1/2) the number  of edges in Hr(n, p) is precisely M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus we have that  �������  χ(Hr(n, M)) −  �  (r−1)d  r log d  �  1  r−1  �  (r−1)d  r log d  �  1  r−1  �������  ≤ ε  2  with probability at least 1 − o(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This proves the first inequality, and the second one  follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  6  Now we will start to transform Hr(n, m) to the random regular hypergraph Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This  transformation will involve first removing some edges from vertices of degree larger than d,  and then adding some edges to vertices of degree less than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We define the rank of a point  pi ∈ Vj, to be the number of points pi′ ∈ Vj such that i′ ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We form a new set of points  P ′ ⊆ P and a partition R′ of P ′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' For any Rk ∈ R containing a point with rank  more than d, we delete Rk from R and delete all points of Rk from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Note that each bucket  contains at most r points of P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Note also that R′ is a uniform random partition of P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We  let HR′ be the natural hypergraph associated with R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Now we would like to put some more points into the buckets until each bucket has exactly d  points, arriving at some set of points P ′′ ⊇ P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We would also like a uniform partition R′′ of  P ′′ into sets of size r, and we would like R′′ to have many of the same parts as R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We will  accomplish this by constructing a sequence P ′  1 := P ′ ⊆ P ′  2 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' ⊆ P ′  ℓ =: P ′′ of point sets  and a sequence R′  1 := R′, R′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' , R′  ℓ =: R′′ where R′  j is a uniform random partition of P ′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We construct P ′  j+1, R′  j+1 from P ′  j, R′  j as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Suppose |R′  j| = a (in other words R′  j has  a parts), so |P ′  j| = ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' P ′  j+1 will simply be P ′  j plus r new points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Now we will choose a  random value K ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' , r} using the distribution P[K = k] = qk(a), where qk(a) is defined  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Consider a random partition of ra+r points into a+1 parts of size r, and fix  some set Q of r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Then for 1 ≤ k ≤ r, the number qk(a) is defined to be the probability  that Q meets exactly k parts of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We will then remove a uniform random set of K − 1 parts from R′  j, leaving Kr points in  P ′  j+1 which are not in any remaining part of R′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We partition those points into K parts of  size r such that each part contains at least one new point (each such partition being equally  likely), arriving at our partition R′  j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We claim that R′  j+1 is a uniform random partition of P ′  j+1 into parts of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Indeed, first  consider the r new points that are in P ′  j+1 which were not in P ′  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' The probability that a  uniform random partition of P ′  j+1 would have exactly k parts containing at least one new  point is qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' So we can generate such a random partition as follows: first choose a random  value K with P[K = k] = qk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' next we choose a uniform random set of (K − 1)r points from  P ′  j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' next we choose a partition of the set of points consisting of P ′  j+1 \\ P ′  j together with the  points from P ′  j we chose in the last step, where the partition we choose is uniformly random  from among all partitions such that each part contains at least one point of P ′  j+1 \\P ′  j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' finally,  we choose a uniform partition of the rest of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' In our case this partition of the rest  of the points comprises the current partition of the “unused” (a − K + 1)r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' At the  end of this process we have that HR′′ is distributed as Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='1  Bounding the number of low degree vertices in HR′  We define some sets of buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We show that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' there are few small buckets i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='e few  vertices of low degree in the hypregraph HR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let S0 be the buckets with at most d−3d1/2 log d  points of P ′, and let S1 be the buckets with at most d−2d1/2 log d points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let S2 be the  set of buckets that, when we remove points from P ′ to get P, have at least d1/2 log d points  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Then S0 ⊆ S1 ∪ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Our goal is to bound the probability that S0 is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Fix a bucket Vj and let X ∼ Bin  �  rm, 1  n  �  be the number of points of P in Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Then the  probability that Vj is in S1 satisfies  P[Vj ∈ S1] = P  �  X ≤ d − 2d1/2 log d  �  = P  �  X − rm  n ≤ −d1/2 log d  �  ≤ exp  �  −  d log2 d  3(d − d1/2 log d)  �  = exp  �  −Ω  �  log2 d  ��  ,  where for our inequality we have used the Chernoff bound (Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Therefore E[|S0|] ≤  exp  �  −Ω  �  log2 d  ��  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Now we argue that |S1| is concentrated using McDiarmid’s inequality  (Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' For our application we let X = |S1| which is a function (say f) of the vector  (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Zrm) where Zi tells us which bucket the ith point of P went into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Moving a point  from one bucket to another can only change |S1| by at most 1 so we use c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus we get  the bound  P(|X − E[X]| ≥ n2/3) ≤ 2 exp  �  − n4/3  2rm  �  = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (12)  Now we handle S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' For 1 ≤ j ≤ n let Yj be the number of parts Rk ∈ R such that Rk  contains a point in the bucket Vj as well as a point in some bucket Vj′ where |Vj′| > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Note  that if Vj ∈ S2 then Yj ≥ d1/2 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We view Rk as a set of r points, say {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' , qr} each  going into a uniform random bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Say qi goes to bucket Vji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' The probability that Rk is  counted by Yj is at most  rP[j1 = j and |Vj1| > d] + r(r − 1)P[j1 = j and |Vj2| > d]  = r  nP[|Vj1| > d  ��j1 = j] + r(r − 1)  n  P[|Vj2| > d  ��j1 = j]  ≤ r2  n P[|Vj1| > d  ��j1 = j]  ≤ r2  n P[Bin(rm − 1, 1/n) ≥ d] = r2  n exp  �  −Ω  �  log2 d  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Thus we have  E[Yj] = m · r2  n exp  �  −Ω  �  log2 d  ��  ≤ rd exp  �  −Ω  �  log2 d  ��  = rd1/2 exp  �  −Ω  �  log2 d  ��  and so Markov’s inequality gives us  P  �  Yj ≥ d1/2 log d  �  ≤ rd exp  �  −Ω  �  log2 d  ��  d1/2 log d  = exp  �  −Ω  �  log2 d  ��  8  and so E[|S2|] = n exp  �  −Ω  �  log2 d  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We use McDiarmid’s inequality once more, this time  with X = |S2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  A change in choice of bucket changes |S2| by at most one and so (12)  continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus  |S0| = n exp  �  −Ω  �  log2 d  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2  A property of independent subsets of Hr(n, m)  Fix 1 ≤ j ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Set  a :=  �  1 + ε  2  � � r log d  (r − 1)d  �  1  r−1  ,  κj := 10d  r  �r  j  �  aj,  p := d(r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  nr−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  The expected number of independent sets A in Hr(n, p) of size at most an such that there  are κjn edges each having j vertices in A is at most  an  �  s=1  �n  s  �  (1 − p)(s  r)  ��s  j  �� n  r−j  �  κjn  �  pκjn  ≤  an  �  s=1  exp  \uf8f1  \uf8f2  \uf8f3s log  �en  s  �  −  �s  r  �  p + κjn log  \uf8eb  \uf8ede(an)j  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  nr−j  (r−j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p  κjn  \uf8f6  \uf8f8  \uf8fc  \uf8fd  \uf8fe  =  an  �  s=1  exp  �  s log  �en  s  �  −  �s  r  �  p + κjn log  �eaj  10  ��  ≤ an · exp  ��  log  �e  a  �  − 10d  r  �r  j  �  aj−1 log  �10  e  ��  an  �  = o(1/n)  where the last line follows since as d → ∞ we have  log  �e  a  �  ∼  1  r − 1 log d  and  10d  r  �r  j  �  aj−1 log  �10  e  �  = Ω  �  d  r−j  j−1 log− j−1  r−1 d  �  ≫ log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Thus with probability 1 − o(1/n), Hr(n, p) has a coloring using (1 + ε/2)  �  (r−1)d  r log d  �  1  r−1 colors  such that for each color class A and for each 1 ≤ j ≤ r − 1 there are at most κjn edges with  j vertices in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' The hypergraph Hr(n, m), m =  �n  r  �  p will have this property w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='.  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='3  Transforming HR′ into Hr(n, d)  Now we will complete the transformation to the random regular hypergraph Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We  are open to the possibility that doing so will render our coloring no longer proper, since this  9  process will involve changing some edges which might then be contained in a color class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We  will keep track of how many such “bad” edges there are and then repair our coloring at the  end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We have to add at most (3d1/2 log d + d exp  �  −Ω  �  log2 d  ��  )n < (4d1/2 log d)n points, which  takes at most as many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  For each color class A of HR′ define XA,j = XA,j(i) to  be the number of edges with j vertices in A at step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We have already established that  XA,j(0) ≤ κjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This follows from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='2 and the fact that we have removed edges  from H(n, m) to obtain HR′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let Ei be the event that at step i we have that for each color  class A and for each 1 ≤ j ≤ r − 1 we have XA,j(i) ≤ 2κjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Then, assuming Ei holds, the  probability that XA,j increases at step i is at most  �  1≤k≤r, jℓ≥1  j1+···+jk=j  �  1≤ℓ≤k  2κjℓn  nd/r =  �  1≤k≤r, jℓ≥1  j1+···+jk=j  �  1≤ℓ≤k  20  � r  jk  �  ajk ≤  �  1≤k≤r, jℓ≥1  j1+···+jk=j  20r2r2aj ≤ 40r2r2aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Also, the largest possible increase in XA,j in one step is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus, the final value of XA,j  after at most (4d1/2 log d)n steps is stochastically dominated by κjn + rY where Y  ∼  Bin  �  (4d1/2 log d)n, 40r2r2aj�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' An easy application of the Chernoff bound tells us  P (Y > 2E[Y ]) ≤ exp(−Ω(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (13)  Note that here  2E[Y ]  κjn  = 8d1/2 log d · 40r2r2ajn  10d  �r  j  �  ajn/r  = O(d−1/2 log d) < 1  for sufficiently large d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus, using (13) and the union bound over all color classes A, we  have w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' the final value of XA,j is at most κjn + 2E[Y ] ≤ 2κjn for all 1 ≤ j ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Now we address “bad” edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' edges contained in a color class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Assuming Ei holds, the ex-  pected number of new edges contained in any color class at step i is at most r(40)r2r2+2rar =  O  �� log d  d  �  r  r−1�  (because it would have to be one of the colors of one of the vertices we are  adding points to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Thus the expected number of bad edges created in (4d1/2 log d)n steps  is stochastically dominated by Z ∼ r · Bin  �  (4d1/2 log d)n, O  ��log d  d  �  r  r−1� �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Another easy  application of Chernoff shows that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Z ≤ 2E[Z] = O(d−1/2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We repair the coloring as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' First we uncolor one vertex from each bad edge, and let  the set of uncolored vertices be U where |U| = u = O  �  d−1/2n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Let  δ := ε  2  �(r − 1)d  r log d  �  1  r−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  We claim that for every S ⊆ U, |S| = s, the hypergraph induced on S has at most δs/r  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This will complete our proof since it implies that the minimum degree is at most δ  and so U can be recolored using a fresh set of δ colors, yielding a coloring of Hr(n, d) using  10  at most  χ(Hr(n, M)) + δ ≤  �  1 + ε  2  � �(r − 1)d  r log d  �  1  r−1  + ε  2  �(r − 1)d  r log d  �  1  r−1  = (1 + ε)  �(r − 1)d  r log d  �  1  r−1  colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' The expected number of sets S with more than δs/r edges is at most  �  1≤s≤u  �n  s  ���ds  r  �  δs/r  �  1  �dn  r  ��dn−r  r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  �dn−δs+r  r  �  ≤  �  1≤s≤u  �ne  s  �s �(dse/r)re  δs/r  �δs/r  (r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' )δs/r  (dn − δs)δs  ≤  �  1≤s≤u  �  ne  s  �  dse  (dn − δs)r  �δ �er · r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  δs  �δ/r�s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  (14)  Now for 1 ≤ s ≤ √n the term in (14) is at most  �  O(n) ·  �  O(n−1/2)  �δ · O(1)  �s  = o(1/n)  since δ can be made arbitrarily large by choosing d large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' Meanwhile for √n ≤ s ≤ u we  have that the term in (14) is at most  �  O(n1/2) · O(1) ·  �  O(n−1/2)  �δ/r�s  = o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  Now since (14) has O(n) terms the whole sum is o(1) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' This completes the  proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  4  Summary  We have asymptotically computed the chromatic number of random r-uniform, d-regular  hypergraphs when proper colorings mean that no edge is mono-chromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' It would seem  likely that the approach we took would extend to other definitions of proper coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content=' We  have not attempted to use second moment calculations to further narrow our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
+page_content='  These would seem to be two natural lines of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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@@ -0,0 +1,1442 @@
+arXiv:2301.01271v1  [econ.GN]  15 Dec 2022
+On the notion of measurable utility on a
+connected and separable topological space:
+an order isomorphism theorem.∗
+Gianmarco Caldini
+7 February 2020
+Abstract
+The aim of this article is to define a notion of cardinal utility function called
+measurable utility and to define it on a connected and separable subset of a weakly
+ordered topological space. The definition is equivalent to the ones given by Frisch
+in 1926 and by Shapley in 1975 and postulates axioms on a set of alternatives that
+allow both to ordinally rank alternatives and to compare their utility differences.
+After a brief review of the philosophy of utilitarianism and the history of utility
+theory, the paper introduces the mathematical framework to represent intensity
+comparisons of utility and proves a list of topological lemmas that will be used in
+the main result. Finally, the article states and proves a representation theorem,
+see Theorem 5, for a measurable utility function defined on a connected and sep-
+arable subset of a weakly ordered topological space equipped with another weak
+order on its cartesian product. Under some assumptions on the order relations,
+Theorem 5 proves existence and uniqueness, up to positive affine transformations,
+of an order isomorphism with the real line.
+∗I am grateful to Professor Massimo Marinacci for letting me know about the open problem.
+1
+
+Introduction
+Together with notions such as value, money, market and economic agents, utility has
+been one of the most controversial concepts in the whole history of economic theory.
+The most important debate can be considered the one around the question whether it
+is possible to define a clear and rigorous concept of utility and an appropriate notion
+of unit of measurement for utility, seen as a quantity like the physical ones. In the first
+chapter we will give a short introduction to the evolution of the concept of utility from
+both a philosophical and a historical point of view. Our treatment is far from being
+exhaustive. For an extensive treatment of history of utility and utility measurements,
+we refer the interested reader to Stigler [39], Majumdar [26], Adams [1], Luce and
+Suppes [24], Fishburn [15], [16], [17] and Moscati [29].
+The second chapter will shift from the descriptive part to more formal concepts
+and will be used to introduce the usual mathematical framework of decision theory.
+Moreover, we will introduce definitions and axioms that will enable us to represent
+comparisons of the intensity that a decision maker feels about the desirability of different
+alternatives. For this aim, we will follow the construction of Suppes and Winet [40]
+and Shapley [36].
+The third and last chapter will be entirely devoted to the proof of Shapley’s theo-
+rem, extending the domain of alternatives X from a convex subset of R to a connected
+and separable subset of a topological space, hence providing a generalization of his
+theorem. Our intended goal is to define a rigorous notion of a specific kind of cardinal
+utility function, not only able to rank alternatives, but also to compare utility differ-
+ences. In particular, we define a “twofold” utility function in line with the primordial
+axiomatization of Frisch [18], calling it a measurable utility function. In mathematical
+terms, we will prove a specific order-isomorphism theorem between a totally ordered,
+connected and separable subset of a topological space and the real line.
+1
+Philosophy and history of utility theory
+Theory of felicity, theory of justice, theory of morality, theory of virtue and theory of
+utility are among the most important theories of moral philosophy and, as such, they
+are constantly sources of questions that often do not find an immediate answer. When
+a human being acts, or when she makes a decision, she is, at the same time, looking for
+1
+
+justifications, either positive or normative, for the decision she has just made. We, as
+human beings, are constantly trying to prove that what we did was the best thing to
+do, in some well-defined sense, or, at least, the less harmful. These justifications take
+into account the means, the ends and all the possible paths we have to reach our goals.
+Moral philosophy is the science that comes into place when we formulate questions
+about the ends, the means and the possible ways to achieve them.
+Moral philosophy is essentially composed by principles, also called norms, on what
+is good and what is bad. They allow to define and to judge human actions, means and
+ends. Sometimes, norms take the form of universal laws to which all human beings
+are subjected.
+Nevertheless, the formulation of moral laws or rules that prescribe
+what a single agent should do or not do are intrinsically tied with history. Historical
+experiences determine our vision of the world. Our moral philosophy is the result of
+different heritages that formed a common culture in which values like human respect, an
+idea of equality between human beings and impartiality are among the most important.
+Together with this general definition of morality, there exist the similar concepts of
+ethics and of role morality - a specific form of professional morality. It was Jeremy Ben-
+tham, in an unfinished manuscript which was posthumously published in 1834, to define
+the neologism deontology in the title of his book Deontology or the Science of Morality.
+The manuscript stated, for the first and only time, the particular aspects of Bentham’s
+utilitarian theory as moral philosophy. This passage is clearly mentioned in Sørensen
+[37]:
+[...] pointing out to each man on each occasion what course of conduct promises to be in
+the highest degree conducive to his happiness: to his own happiness, first and last; to the
+happiness of others, no farther than in so far as his happiness is promoted by promoting
+theirs, than his interest coincides with theirs (p. 5).
+In this passage we can see how Bentham considered deontology to be primarily
+aimed at one’s own private felicity. Nevertheless, this does not bring any selfish concern.
+Bentham’s goal can be identified with the objective study and measurement of passions
+and feelings, pleasures and pains, will and action. Among these particular pleasures are
+those stemming from sympathy - in Adam Smith’s sense - and they include the genuine
+pleasure being happy for the good of others.
+In this light, Bentham spent his life in search of the cardinal principle of ethics
+and he found it in Epicurean ethics of hedonism. Hedonism comes from Greek ῾ηδον´η,
+which means pleasure. Thus, classic utilitarianism, founded on hedonism, started from
+2
+
+the principle that pleasure is an intrinsic positive value and sorrow is an intrinsic neg-
+ative value. It is, for this reason, somehow curious that Bentham conception, founded
+on pleasure, had been called utilitarianism, from the simple observation that what is
+useful is not necessarily pleasant or providing pleasure. We need always to take into
+account that the term utility is intended in a functional sense; what gives utility is what
+contributes the most to the individual, or universal, pleasure.
+Classical utilitarian philosophers considered utilitarianism well-founded and realistic
+thanks to the fact that it is based on pleasure. It is well-founded as its norms are jus-
+tified by an intrinsic, absolute value, that does not need any further justification. It is
+realistic because they thought human being to ultimately seek the maximum pleasure
+and the minimum sorrow. More specifically, human beings try to choose the action
+that will provide the maximum excess of pleasure against grief.
+For Bentham, what really matters is the total amount of pleasure, intended as
+the total excess of pleasure against sorrow: the only reasons for human actions are the
+quests for pleasure, avoiding sorrow: they are the sources of our ideas, our judgments
+and our determinations. Human moral judgments become statements on happiness;
+pleasure (or felicity) is good and sorrow is bad. Utilitarian moral can be considered as
+a “calculated hedonism”, that carefully evaluates the characteristics of pleasure. Wise
+is the man that is able to restrain from an immediate pleasure for a future good that,
+in comparison, will be more beneficial. On the other side, being able to evaluate the
+positive or negative consequences of an action without making mistakes is fundamental.
+Hence, the correct utilitarian person should reach some kind of “moral arithmetic” that
+allows the correct calculations to be carried out. Far from being a unanimously accepted
+doctrine, we cannot forget to mention that Alessandro Manzoni wrote an essay [27] in
+which he strongly criticized Bentham’s utilitarianism, saying that it is utterly wrong to
+think that human beings build their moral values judgment of their actions on utility.1
+From this explanation of utilitarianism, Bentham’s evaluation criterion of actions
+follows as an immediate corollary: the maximum happiness for the maximum number
+of people. Again, happiness is intended as state of pleasure, or absence of grief. Hence,
+individual pleasure becomes no more the ultimate goal: it is the universal pleasure to
+be hegemonic.2
+1Manzoni [27] wrote: ”Non ci vuol molto a scoprir qui un falso ragionamento fondato sull’alterazione
+d’un fatto. Altro `e che l’utilit`a sia un motivo, cio`e uno de’ motivi per cui gli uomini si determinano
+nella scelta dell’azioni, altro `e che sia, per tutti gli uomini, il motivo per eccellenza, l’unico motivo
+delle loro determinazioni (p.775).
+2This tension between individual pleasure and universal pleasure is one of the principal difficulties
+3
+
+This view of utilitarianism admits, at least in our minds, the conception of the
+existence of a scale of pleasure in which pleasure and sorrow can be added and sub-
+tracted. In other words, the idea of a calculus of felicity and grief is not completely
+absurd, both in intrapersonal and interpersonal compensations.
+1.1
+Brief history of utilitarianism
+Although it is possible to find utilitarian reasonings in Aristotele’s works, it is com-
+monly agreed that the beginning of the history of utility can be identified with 18th
+century moral philosophy. To be even more specific, Bentham’s ideas were not isolated,
+since they were already present in works by his illuministic predecessors like Richard
+Cumberland, Francis Hutcheson and Cesare Beccaria. Especially Hutcheson [20] had
+already defined good as pleasure and good objects as objects that create pleasure. The
+novelty of Bentham was to treat pleasure as a measurable quantity, thus making the
+utilitarian doctrine directly applicable to issues like tax policies and legislation. In-
+deed, not only did Bentham argue that individual pleasure was measurable, but also
+that happiness of different people could be compared. Stark [38] cited in his article
+Bentham’s writings in the following way:
+Fortunes unequal: by a particle of wealth, if added to him who has least, more happiness
+will be produced, than if added to the wealth of him who has most (vol. 1, p. 103).
+Stark [38] continues:
+The quantity of happiness produced by a particle of wealth (each particle being the same
+magnitude) will be less and less every particle (vol. 1, p. 113).
+It is easy to see how this last concept and the well-known idea of decreasing
+marginal utility are related.
+In the pioneering work of Jevons [21], utility functions were the primitive mathe-
+matical notion to formalize and quantify Bentham’s calculus of pleasure. Utility func-
+tions were tools to measure and scale the amount of well-being of human beings. It
+seems clear, at this point, how the starting role of utility functions was cardinal,3 in the
+sense that utility or, better, pleasure differences were well-founded and realistic notions
+with a strong moral philosophy justification.
+of utilitarian moral philosophy.
+3Note that before the work of Hicks and Allen [19], economists spoke about measurable utility and
+not of cardinal utility.
+4
+
+Summing up, in the beginning, utility functions were designed for the mere purpose
+of a calculus of pleasure and sorrow. However, even if the philosophical concept made
+sense, the difficulties in the quantification of any experimental measurement of pleasure
+led cardinal utility theory to be seen more just like a thought process rather than a
+science.
+However, utility theory did not rise from philosophy alone, but it was object of
+study of other sciences such as statistics, with the so-called St. Petersburg paradox,
+and psychophysics, the study of physical stimuli and their relation to sensory reactions.
+These two phenomena can be considered the starting point of the law of decreasing
+marginal utility. It was Nicolas Bernoulli that, originally, invented what is now called
+the St. Petersburg puzzle, which offered the theoretical explanation for the law of de-
+creasing marginal utility of wealth. The standard version of the puzzle is the following:
+a fair coin is tossed until it lands “head” on the ground. At that point, the player wins
+2n dollars, where n is the number of times the coin was flipped. How much should one
+be willing to pay for playing this game? In other words, what is the expected value of
+the game, given the probability of “head” being 0.5? The mathematical answer is
+∞
+�
+i=1
+1
+2i · 2i = 1 + 1 + · · · = ∞.
+The only rationale for this conundrum is that, if it makes sense to maximize expected
+utility and if people are willing to participate to the St. Petersburg game for only a
+finite amount of money, then their marginal utility as a function of wealth must be,
+somewhere, decreasing.
+Neither Bentham nor Bernoulli thought as decreasing marginal utility as a phe-
+nomenon in need of scientific justifications. Nevertheless, this came as an immediate
+consequence from the psychophysical theories discovered by Weber [42] and generalized
+by Fechner [13]. One of the most important questions posed by psychophysics is what is
+the functional link between different degrees of a given stimulus and a given sensation.
+What Weber did was an in-depth study to try to measure the smallest detectable change
+(also called “just noticeable difference” or “minimum perceptible threshold”) in stimuli
+like heat, weight and pitch. Moreover, Fechner took this “just noticeable difference”
+as a unit of measurement, constructing a scale for subjective sensations. From their
+studies we now have the so-called Weber’s law and Fechner’s laws: the former states
+that the relative increase of a stimulus needed to produce a just noticeable change is
+5
+
+constant and, the latter, that the magnitude of sensation is a logarithmic function of
+the stimulus.
+In conclusion, if wealth is a stimulus, then Benthamian utility must be the cor-
+responding sensation.
+In this light, St.
+Petersburg puzzle can be seen as just one
+materialization of these laws.
+At the end of 19th century, the marginalist revolution paved the way for an ordinal
+approach to the notion of utility. In fact, this was because one of the main economic
+problems of late 19th century was the need of a theory of demand. One of the leading
+figures that founded neoclassical theory with scientific and analytic rigor was Vilfredo
+Pareto. Pareto is considered the father of the so-called ordinal approach. It was a notion
+of utility that was purely comparative and it left out from the theory the initial idea for
+which utility theory was developed: the existence of psychophysical and physiological
+substrates. Pareto’s theory was so successful that was considered a revolution in the
+notion of utility. The ordinal approach was extremely successful because it solved the
+classic consumer problem based on indifference curves, and the notion of utility had a
+central role in its construction. The key aspect was the replacement of marginal utility
+- a notion that was meaningless in an ordinal approach - with the trick of marginal rate
+of substitutions along indifference curves.
+Interesting are the writings of Francis Edgeworth [10] and Pareto [31], starting
+from very different assumptions and arriving at different conclusions.
+Edgeworth’s
+main contribution can be summarized in the synthesis of Bentham’s utilitarianism and
+Fechner’s psychophysics: his ideas were based on the unit of utility seen as a just
+perceivable increment of pleasure. Moreover, he was interested also in an inter-personal
+unit of utility to be able to carry out welfare comparisons among people. Edgeworth
+was completely aware of the impossibility of testing these implications, but he was a
+strong supporter of the idea of possible comparisons of happiness among people.
+Pareto, on the other hand, denied Edgeworth’s intuition of comparisons of utility.
+Instead, Pareto [31] reckoned the theoretical possibility of a cardinal notion of utility,
+seen as the limit of the purely comparative notion he developed. Nevertheless, he also
+argued that such a notion of perfect precision is not attainable and that pleasure is only
+imperfectly measurable.4 Summing up, Edgeworth’s and Pareto’s ways of conceiving
+measurable utility must be differentiated and utility theory is still today based on the
+Paretian notion mainly because of its use in the theory of demand and in the general
+4However, Pareto [32] writes: “There is no reason for not accepting it [cardinal utility], with the
+reservation that it must be verified by the results deduced from it (p. 73).”
+6
+
+equilibrium theory.
+In 1950, ordinalism was the well-established mainstream ideology in utility theory
+and the cardinal notion of utility was almost completely abandoned. Nevertheless, the
+purely comparative approach was not convincing everyone, mainly because people’s
+introspection suggested the existence something more. One of the main supporters of
+cardinalism was Maurice Allais who explicitly wrote in [4]:
+The concept of cardinal utility [...] has almost been rejected the literature for half a cen-
+tury. This rejection, based on totally unjustified prejudices, deprived economic analysis
+of an indispensable tool (p. 1).
+Allais [4] admits that the theory of general economic equilibrium can be fully de-
+scribed in an ordinal world, but he immediately lists a series of theories that cannot
+be adequately developed without a rigorous and well-defined concept of cardinal utility
+and interpersonal comparisons. Some examples are the theory of dynamic evolution of
+the economy, the theory of fiscal policy, of income transfers, of collective preferences,
+social welfare analysis and political choices, of risk, of insurance and the theory of
+cooperative games. Then, Allais goes even further in his defense of cardinal utility,
+arguing that even the theory of demand could become more intuitive - and with a
+simpler exposition - if we could appeal to a notion of intensity of preferences. In any
+case, as long as the conclusions of price theory do not change significantly using the
+ordinal and the cardinal approach, we should prefer the purely comparative approach
+by Occam’s razor. But the problems with group decision making, social choice theory
+and cooperative game theory still cannot be solved. Indeed, while classical economists
+considered distributional problems as a fundamental part of economic science, the ordi-
+nalist approach to utility theory refused completely to deal with questions that involved
+interpersonal comparisons of welfare. Economists became more interested in positive
+statements, rather than normative ones and the accent was put on efficiency, rather
+than equity. This was the case of the optimum allocations in the sense of Pareto. For
+a complete overview of the main issues of welfare economics, the main problems with
+an ordinal approach and the main literature, we refer the interested reader to Sen [35].
+Hence, one of the main problems, still in the 21st century, is how is it possi-
+ble to understand the intuitive tool of introspection to develop a rigorous theory that
+economists can apply in their models and explain economic phenomena. The solution
+does not exists yet. In the last years, the issue started getting the attention of few deci-
+sion theorists mainly because of the powerful developments in the field of neuroscience
+7
+
+and the new discipline of neuroeconomics. These fields of cognitive sciences are going
+into the direction of overcoming the main difficulty of the founders of utilitarianism:
+the difficulty of carrying out experiments on pleasure and pain and the construction of
+a rigorous and well-defined scale of pleasure. Nevertheless, nothing is clear yet, mainly
+because also the theoretical concept of cardinal utility is still vague. Cardinal utility
+is still used as a name for a large number of formally distinct concepts and it misses a
+precise and well-established definition that can be applied in decision-theoretic models.
+During the 20th century a lot of methodologies to try to define a concept of
+measurement of human sensation have been defined.
+Definition 1. A scale is a rule for the assignment of numbers to aspects5 of objects or
+events.
+The result was the development of a full taxonomy of scales, with scales that differ
+in terms of higher precision of measurement. For an extensive treatment of the theory
+of measurement we refer the interested reader to Krantz et alii [22].
+The issue of having a rigorous definition for cardinal utility was not solved by the
+theory of measurement.
+It was just translated in a different language: what is the
+suitable scale for measuring a given aspect? The definition of a unit of measurement
+for utility was not an easy task to solve. Even in physics, where experiments can be
+carried out with relatively high precision, the way a unit of measurement is defined
+is not perfect. One meter was originated as the 1/10-millionth of the distance from
+the equator to the north pole along a meridian through Paris. Then, the International
+Bureau of Weights and Measures, founded in 1875, defined the meter as the distance
+of a particular bar made by platinum and iridium kept in S`evres, near Paris. More
+recently, in 1983, the Geneva Conference on Weights and Measures defined the meter
+as the distance light travels, in a vacuum, in 1/299,792,458 seconds with time measured
+by a cesium-133 atomic clock which emits pulses of radiation at very rapid and regular
+intervals.
+Increases in science allow the unit of measurement to be duplicated with a better
+and better level of precision. The comparison with the unit of measurement of the
+quantity utility can be carried out with the philosophical question whether it is, for
+some esoteric reason, intrinsically impossible to measure human beings’ pleasure or
+whether economic science and neuroeconomics are so underdeveloped that we still have
+5For example: hardness, length, volume, density, . . .
+8
+
+very poor precision in measuring human felicity.6
+The same comparison can be done with light (or heat, color and wave lengths, as
+it is mentioned by von Neumann and Morgenstern in [41]). For example, temperature
+was, in the original concept, an ordinal quantity as long as the concept warmer was
+known. Then, the first transition can be identified with the development of a more pre-
+cise science of measurement: thermometry. With thermometry, a scale of temperature
+that was unique up to linear transformations was constructed. The main feature was
+the association of different temperatures with different classes of systems in thermal
+equilibrium. Classes like these were called fixed points for the scale of temperatures.
+Then, the second transition can be associated with the development of thermodynam-
+ics, where the absolute zero was fixed, defining a reference point for the whole scale.
+In physics, these phenomena had to be measured and the individual had to be able to
+replicate results of such measurements every time. The same may apply to decision
+theory and the notion of utility, someday. At the moment, the issue remains unclear,
+even if even Pareto was not completely skeptical about the first transition from an
+ordinal purely comparative approach to that of an equality relation for utility differ-
+ences. Von Neumann and Morgenstern point out in [41] that the previous concept is
+based on the same idea used by Euclid to describe the position on a line: the ordinal
+utility concept of preference corresponds to Euclid’s notion of lying to the right of and
+the derived concept of equality of utility differences with the geometrical congruence of
+intervals.
+Hence, the main question becomes whether the derived order relation on utility
+differences can be observed and reproduced. Nobody can, at the moment, answer this
+question.
+1.2
+Axiomatization of utility theories
+In 1900, at the International Congress of Mathematicians in Paris, David Hilbert an-
+nounced that he was firmly convinced that the foundation of mathematics was almost
+complete. Then, he listed 23 problems to be solved and to give full consistency to
+mathematics. All the rest was considered, by him, just details. Some of the problems
+6Some authors, like Ellingsen [12], are certain, instead, that the philosophical question of whether
+utility is intrinsically measurable or not is a spurious one, mainly because they see the issue of “mea-
+surement” as a concept that is always invented and never discovered. In this light, our question can
+be rephrased as whether it is possible to define a correct notion of measurement that allows some kind
+of intrapersonal and interpersonal utility comparisons.
+9
+
+consisted in the axiomatizations of some fields of mathematics. Indeed, at the begin-
+ning of the 20th century, the idea of being able to solve every mathematical problem
+led mathematicians to try develop all mathematical theory from a finite set of axioms.
+The main advantage of the axiomatic method was to give a clean order and to remove
+ambiguity to the theory as a whole. Axioms are the fundamental truths by which it
+is possible to start modeling a theory. The careful definition of them is critical in the
+development of a theory that does not contain contradictions.
+As a result, almost all fields of science started a process of axiomatization, utility
+theory as well.
+The ordinal Paretian revolution was the fertile environment where
+preferences started to be seen as primitive notions. Preference relations began to be
+formalized as mathematical order relations on a set of alternatives X and became the
+starting point of the whole theory of choice.
+As a result, utility functions became
+the derived object from the preference relations. The mainstream notion of ordinal
+(Paretian) utility reached its maturity with the representation theorems by Eilenberg
+[11] and Debreu [8], [9]. Subsequent work in decision theory shifted from decision theory
+under certainty to choice problems under uncertainty, with the pioneering article of
+Ramsey [33] on the “logic of partial belief.” In short, Ramsey [33] stated the necessity
+of the development of a purely psychological method of measuring both probability
+and beliefs, in strong contradiction with Keynes’ probability theory. Some years after,
+the milestone works of von Neumann and Morgenstern [41] and Savage [34] gave full
+authority to decision theory under uncertainty.
+One of the first treatments of preference relations as a primitive notion can be
+identified with Frisch [18], in his 1926 paper.
+Ragnar Frisch was also the first to
+formulate an axiomatic notion of utility difference. Hence, two kinds of axioms were
+postulated by him: the first ones - called “axioms of the first kind” - regarded the
+relation able to rank alternatives in a purely comparative way, while the second axioms
+- named “axioms of the second kind” - reflected a notion of intensity of preference and
+allowed utility differences to be compared.
+So, in parallel to the axiomatization of
+ordinal utility, also cardinal utility axiomatizations started to grow.
+Frisch’s article did not have the deserved impact in the academic arena, mainly
+because his article was written in French and published in a Norwegian mathematical
+journal. Hence, the full mathematical formalization of these two notions of preference
+axioms resulted almost ten years later from the 1930s debate by Oskar Lange [23] and
+Franz Alt [5]. Lange [23] defined an order relation ≻ on the set of alternatives X with
+the meaning that, for any two alternatives x, y ∈ X, x ≻ y reads “x is strictly preferred
+10
+
+to y.” Then, a corresponding relation P on ordering differences is assumed with the
+meaning that, for any x, y, z, w ∈ X, xyPzw reads “a change from y to x is strictly
+preferred than a change from w to z.
+More formally:
+x ≻ y ⇐⇒ u(x) > u(y) for all x, y ∈ X
+(1)
+xyPzw ⇐⇒ u(x) − u(y) > u(z) − u(w) for all x, y, z, w ∈ X
+(2)
+The main theorem of Lange [23] can be stated as follows:
+Theorem 1. If there exists a differentiable utility function u : R → R such that (1)
+and (2) hold, then only positive affine transformations of that utility function represent
+the given preferences ≻ and P.
+It is immediate to see that Lange [23] provides only necessary conditions for a
+utility function representation of preference relation. Moreover, it is relatively easy to
+see that the assumption of differentiability of u can be largely relaxed. Hence, the issue
+becomes whether it is possible to find sufficient conditions on the preference relations
+under which Lange’s utility function - a cardinal utility function - exists. This was
+done by Franz Alt in his 1936 article [5]. Alt postulated seven axioms that guaranteed
+sufficient and necessary conditions for the existence of a continuous utility function
+- unique up to positive affine transformations - based on a preference relation and a
+utility-difference ordering relation. In his set of axioms, Alt defined a notion that can
+be understood as the set of alternatives X to be connected.
+With Frisch’s pioneering work of 1926 and 1930s debate by Lange and Alt, the
+modern ingredients of cardinal utility axiomatization such as equations (1) and (2) and
+connectedness of the domain of alternatives X started to be formalized. In those years,
+a lot of different axiomatic models were studied, till the article of the famous philoso-
+pher of science Patrick Suppes and his doctoral student Muriel Winet [40]. In their
+1955 paper, Suppes and Winet developed an abstract algebraic structure of axioms for
+cardinal utility, called a difference structure, in line with old Frisch’s ideas and Lange’s
+formalization: not only are individuals able to ordinally rank different alternatives, but
+they are also able to compare and rank utility differences of alternatives. Indeed, Sup-
+pes and Winet cited the work of Oskar Lange on the notion of utility differences and
+stood in favor of the intuitive notion of introspection, elevating it to not just a mere
+11
+
+intuition, but as a solid base where to build a notion of utility differences. Suppes and
+Winet continued their article saying that, up to 1950s, no adequate axiomatization for
+intensity comparison had been given. Hence, as Moscati [29] nicely highlights, they
+were probably unaware of Alt’s representation theorem and this was probably due to
+the fact that Alt [5] was published in German in a German journal. Suppes and Winet
+postulated 11 axioms in total, some on the set of alternatives X and others on the
+two order relations,7 providing sufficient and necessary condition for a cardinal utility
+representation, unique up to positive affine transformations. Another approach to the
+field of axiomatization of cardinal utility was taken twenty years later by Lloyd Shap-
+ley. While axiomatizations `a la Suppes and Winet started developing a set-theoretic
+abstract structure, Shapley substituted the usual long list of postulates with strong
+topological conditions both on the domain of alternatives X and on the topology in-
+duced by the order relations. Shapley [36] constructed a cardinal utility function u
+satisfying some consistency axioms between the orders and assuming the domain of u
+to be a convex subset of the real line. We will enter into the details of Shapley [36] in
+the next chapters.
+In conclusion, the notion of cardinal utility has always suffered a lack of conceptual
+precision in its whole history and, for some authors like Ellingsen [12], it can be even
+considered the main reason why scientists have disagreed over whether pleasure can be
+measured or not.8 What is certain is that the history of cardinal utility, a part from some
+sporadic articles, has been a persistent failure, mostly in its applications to economic
+theory. While the main reason can be probably identified with the almost total absence
+of any rigorous and proven experimental measurement of pleasure, it is fair to observe
+that part of its failure must be given to the strong reluctant opinion of the mainstream
+ordinal “party.” In fact, a large class of economists classify as “meaningless” even the
+mere introspective idea of a comparison of utility difference, and not just the concept
+itself, when formalized in a purely comparative environment. This position is shown to
+be, with a gentle expression, “epistemological laziness.” We should always remember
+that no real progress in economic science can be derived from purely abstract reasoning,
+but only from the combined effort of empirical measurements with theoretical analysis,
+always under the wise guide of the compass of history and philosophy.
+7The conditions these axioms impose are analogous to the conditions defined by Alt [5]: com-
+pleteness, transitivity, continuity, and some form of additivity for the two order relations, and an
+Archimedean property on the quaternary relation.
+8Ellingsen [12] writes about a “fallacy of identity” and “fallacy of unrelatedness.”
+12
+
+2
+Preliminary results
+The aim of this chapter is twofold. On one side, we introduce the mathematical frame-
+work that enable us to represent intensity comparisons that a decision maker feels about
+the desirability of different alternatives. For this aim, we follow the construction of Sup-
+pes and Winet [40] and Shapley [36]. On the other side, we state and prove a list of
+lemmas that will be used in Theorem 5 and that allow us to generalize Shapley’s proof
+to a connected and separable subset of a topological space.
+2.1
+Basic definitions
+Definition 2. A relation on a set X is a subset ≿ of the cartesian product X × X,
+where x ≿ y means (x, y) ∈ ≿.
+In decision theory, ≿ is usually called a preference relation, with the interpretation
+that, for any two elements x, y ∈ X, we write x ≿ y if a decision maker either strictly
+prefers x to y or is indifferent between the two.
+Definition 3. An equivalence relation on a set X is a relation R on X that satisfies
+1) Reflexivity: for all x ∈ X, we have xRx.
+2) Symmetry: for any two elements x, y ∈ X, if xRy, then yRx.
+3) Transitivity: for any three elements x, y and z ∈ X, if xRy and yRz, then xRz.
+Definition 4. A relation ≿ on a set X is called a total order relation (or a simple order,
+or a linear order) if it has the following properties:
+1) Completeness: for any two elements x, y ∈ X, either x ≿ y or y ≿ x or both.
+2) Antisymmetry: for any two elements x, y ∈ X, if x ≿ y and y ≿ x, then x = y.
+3) Transitivity: for any three elements x, y and z ∈ X, if x ≿ y and y ≿ z, then
+x ≿ z.
+Note that if ≿ is complete, then it is also reflexive. The relation ≿ induces, in
+turns, two other relations. Specifically, for any two elements x, y ∈ X we write:
+(i) x ≻ y if x ≿ y but not y ≿ x.
+13
+
+(ii) x ∼ y if x ≿ y and y ≿ x.
+It is easy to see, indeed, that if ≿ is reflexive and transitive, then ∼ is an equivalence
+relation. Given an equivalence relation ∼ on a set X and an element x ∈ X, we define
+a subset E of X, called the equivalence class determined by x, by the equation
+E := {y ∈ X : y ∼ x}
+Note that the equivalence class E determined by x contains x, since x ∼ x, hence
+E is usually denoted as [x].
+We will denote X/∼ the collection {[x] : x ∈ X} of
+all equivalence classes, which is a partition of X: each x ∈ X belongs to one, and
+only one, equivalence class. In decision theory, an equivalence class is often called an
+indifference curve.
+Definition 5. A relation ≿ on a set X is called a weak order if it is complete and
+transitive.
+The problem of finding a numerical representation for a preference relation ≿, i.e.
+an order isomorphism between a generic set X and R, has been widely studied by math-
+ematicians and is a familiar and well-understood concept. Such an order isomorphism
+is called, in decision theory, a utility function. More formally:
+Definition 6. A real-valued function u : X → R is a (Paretian) utility function for ≿
+if for all x, y ∈ X we have
+x ≿ y ⇐⇒ u(x) ≥ u(y)
+Utility functions “shift” the pairwise comparisons that characterize the order rela-
+tion ≿ and its properties in the more analytically convenient space of the real numbers.
+Nevertheless, as a result, the only thing that is preserved is the order, and the real
+numbers that are images of the utility function cannot be interpreted as a scale where
+the decision maker can compare different intensities about the single desirability of any
+two alternatives x, y ∈ X. What is important is the ranking given by the real num-
+bers, according to the usual order of the ordered field (R, ≥). Indeed, one can easily
+prove that every strictly increasing transformation of a utility function is again a utility
+function. For this reason, utility functions are called ordinal and their study belong
+to what is called ordinal utility theory. The main problem of ordinal utility theory is
+to study sufficient and necessary conditions under which a relation ≿ admits a utility
+representation. The original reference can be identified with Cantor [7], but the result
+has been adapted by Debreu [8].
+14
+
+In addition, to be able to solve optimization problems, one of the properties that
+is desirable to have is continuity of the utility function. Debreu [8] is the first to state
+the theorem in the way we are going to. Nevertheless, he proved it making explicit
+reference to Eilenberg [11]. We state here a version of this very well-known theorem.
+Definition 7. A weak order ≿ on a set X is said to be continuous if, for every y ∈ X,
+the sets {x ∈ X : x ≻ y} and {x ∈ X : x ≺ y} are open.9
+Theorem 2 (Eilenberg). Let ≿ be a complete and transitive relation on a connected
+and separable topological space X. The following conditions are equivalent:
+(i) ≿ is continuous.
+(ii) ≿ admits a continuous utility function u : X → R.
+One of the biggest theoretical problems of ordinal utility theory is that the expres-
+sion
+u(x) − u(y)
+is a well-defined real number thanks to the algebraic properties of R, but it is meaning-
+less in term of the interpretation of a difference of utility of two alternatives x, y ∈ X.
+In other words, a Paretian utility function does not have an intrinsic introspective psy-
+chological notion of intensity of the preferences. An immediate corollary of this remark
+is that the concept of marginal utility (and what is known under the Gossen’s law of
+decreasing marginal utility), based on the notion of different quotient, is meaningless.
+More formally, the expression
+du(x)
+dx
+= lim
+h→0
+u(x + h) − u(x)
+h
+has no meaning in this setting. Nevertheless, the concept of marginal utility has been a
+milestone in economic theory, proving that this notion deserves an adequate theoretical
+foundation.
+2.2
+An overview on measurable utility theory
+Let X be a set of alternatives. Pairs of alternatives (x, y) ∈ X × X are intended to
+represent the prospect of replacing alternative y by alternative x, that can be read as
+9Note that this is the usual order topology on X.
+15
+
+“x in lieu of y”. Define the binary relation ≽ on X ×X called intensity preference with
+the following interpretation: for any two pairs (x, y) and (z, w) in X × X,
+(x, y) ≽ (z, w)
+is intended to mean that getting x over y gives at least as much added utility as getting
+z over w or (if y ≿ x) at most as much added sadness. As a result, our decision maker
+is endowed with a weak order preference relation ≿ on alternatives and an intensity
+preference relation ≽ on pairs of alternatives.
+Shapley [36] proves his theorem assuming X to be a convex subset of R. As a
+result, the proof exploits the full algebraic power of the ordered field and the topological
+properties of the linear continuum. Our aim is to generalize the set of alternatives X
+to a connected and separable subset of a topological space, ordered with the binary
+relations ≿ and ≽ and with the order topology induced by the weak order ≿.
+We assume the following axioms for ≿ and ≽, as in Shapley [36].
+Axiom 1. For all x, y, z ∈ X we have (x, z) ≽ (y, z) ⇐⇒ x ≿ y.
+Axiom 1 (henceforth A1) is an assumption of consistency between the two order-
+ings because it implies that the decision maker prefers to exchange z with x instead of
+z with y if and only if she prefers x to y. Together with A1 we can formulate a dual
+version of consistency, A1′, that can be derived from the whole set of axioms we are
+going to assume later.10
+Axiom 1′. For all x, y, z ∈ X we have (z, x) ≽ (z, y) ⇐⇒ y ≿ x.
+We now introduce the main object of this thesis: a joint real-valued representation
+for the two orders ≿ and ≽.
+Definition 8. A real-valued function u : X → R is a measurable utility function for
+(≿, ≽) if for each pair x, y ∈ X
+x ≿ y ⇐⇒ u(x) ≥ u(y)
+(3)
+and if, for each quadruple x, y, z, w ∈ X
+(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w).
+(4)
+The measurable terminology has nothing to do with measure theory, but it refers to
+what is known as measurement theory, i.e. the field of science that established the for-
+mal foundation of quantitative measurement and the assignment of numbers to objects
+10We mention A1′ as a form of axiom only because in this way we can refer to it in the proof of
+Theorem 5, but we never assume it formally. A proof of it will be formulated forward with Lemma 13.
+16
+
+in their structural correspondence. Indeed, not only is a measurable utility function
+able to rank pairs of alternatives according to a preference relation, but it also repre-
+sents the idea of magnitude and intensity of the preference relation among alternatives.
+Therefore, the numerical value u(x) that a measurable utility function assigns to the
+alternative x is assuming the role of a particular unit of measurement for pleasure, that
+we call util.
+Recall that an ordinal utility function u is unique up to strictly monotone trans-
+formations f : Im(u) → R. Hence, a measurable utility function is not ordinal. Never-
+theless, it is unique up to positive affine transformations. Recall that a positive affine
+transformation is a special case of a strictly monotone transformation of the follow-
+ing form f(x) = αx + β, with α > 0 and β ∈ R. Positive affine transformations are
+order-preserving thanks to α > 0.
+Proposition 1. A measurable utility function u : X → R for (≿, ≽) is unique up to
+positive affine transformations.
+Proof. If u(x) = αu(x) + β then we have
+x ≿ y ⇐⇒ u(x) ≥ u(y) ⇐⇒ u(x) = αu(x) + β ≥ αu(y) + β = u(y)
+and
+(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w)
+⇐⇒ u(x) − u(y) = α[u(x) − u(y)] ≥ α[u(z) − u(w)] = u(z) − u(w).
+As a result, u and u are two utility representations for (≿, ≽).
+The whole class
+of utility functions that are unique up to positive affine transformations are called
+cardinal. Measurable utility functions are, therefore, cardinal and pertain to the so-
+called cardinal utility theory.
+Other two axioms (A2, A3) we need to introduce are the following:
+Axiom 2. For all x, y, z, w ∈ X we have (x, y) ∼ (z, w) ⇐⇒ (x, z) ∼ (y, w).
+Axiom 3. For all x, y, z, w ∈ X the set
+{(x, y, z, w) ∈ X × X × X × X : (x, y) ≽ (z, w)}
+is closed in the product topology.
+Axiom 2 is a “crossover” property that characterizes difference comparisons of util-
+ity, while Axiom 3 is a technical assumption defining the order relation ≽ as continuous.
+17
+
+Shapley [36] proves his theorem on a domain of alternative outcomes that is a
+nonempty, convex subset D of the real line where the preference order coincides with
+the total order of (R, ≥). Moreover, ≽ is assumed to be a weak order on D × D such
+that A1, A2 and A3 are satisfied.
+Theorem 3 (Shapley). There exist a utility function u : D ⊆ R → R such that
+x ≥ y ⇐⇒ u(x) ≥ u(y)
+(5)
+and
+(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w)
+(6)
+for all x, y, z, w ∈ D. Moreover, this function is unique up to a positive affine transfor-
+mation.
+The theorem is stated as a sufficient condition, which is the most difficult part to
+prove. The necessary condition of the theorem is easily proved and we state it here as
+a proposition.
+Proposition 2. If the pair (≥, ≽) has a continuous measurable utility function u : D ⊆
+R → R, then ≥ is complete and transitive, ≽ is complete, transitive, continuous (A3)
+and satisfies the crossover axiom (A2), and jointly ≥ and ≽ satisfy the consistency
+axiom (A1).
+Shapley’s construction of the measurable utility function of Theorem 3 is extremely
+elegant, but has the drawback of being too specific as u is defined on a convex subset
+of R. On the other side of the spectrum, as mentioned in the first chapter, the field
+of utility axiomatization has been prolific in the 20th century and a copious number
+of cardinal-utility derivations from preference-intensity axiomatizations were published.
+One of the most important papers on this issue was the one published in 1955 by Patrick
+Suppes and Muriel Winet. Recalling what described before, Suppes and Winet [40]
+advanced an axiomatization of cardinal utility based on the assumption that individuals
+are not only able to rank the utility of different alternatives, as is assumed in the ordinal
+approach to utility, but are also capable of ranking the differences between the utilities
+of commodities. Nevertheless, their 11 axioms on an abstract algebraic structure were
+not fully satisfactory in terms of generality: it was too general. Indeed, some of their
+axioms can be derived in Shapley [36], thanks to the topological properties of R.
+The aim of this research is to settle somewhere in between, finding a representation
+theorem for cardinal utility function (in particular, a measurable one) keeping the
+18
+
+elegance of Shapley’s proof and generalizing the domain of alternatives into the direction
+of Suppes and Winet [40]. We will state and prove a representation theorem for a
+measurable utility function u : X → R where X is a connected and separable subset of
+a topological space, ≿ and ≽ are weak orders and they satisfy (A1), (A2) and (A3).
+Before doing this, we need to state and prove some topological preliminary results that
+will be used in Theorem 5.11
+2.3
+A few basic lemmas
+Definition 9. Let X be a topological space. X is connected if it cannot be separated
+into the union of two disjoint nonempty open subsets. Otherwise, such a pair of open
+sets is called a separation of X.
+Definition 10. Let X be a topological space. X is separable if there exists a countable
+dense subset. A dense subset D of a space X is a subset such that its closure equals the
+whole space, i.e. D = X.
+Definition 11. A totally ordered set (L, ≿) having more than one element is called a
+linear continuum if the following hold:
+(a)
+L has the least upper bound property.
+(b)
+If x ≻ y, there exists z such that x ≻ z ≻ y
+We recall that a ray is a set of the following type (−∞, a) = {x ∈ L : x ≺ a}
+and (−∞, a] = {x ∈ L : x ≾ a} in the case L does not have a minimum. In the
+case L does have a minimum we write [xm, a) = {x ∈ L : xm ≾ x ≺ a} and [xm, a] =
+{x ∈ L : xm ≾ x ≾ a}. Analogously for the sets (a, +∞), [a, +∞), (a, xM] , [a, xM], where
+xM is the maximum of L in the case it existed.12
+Given A ⊆ X, an element y ∈ X is an upper bound for a set A if y ≿ x for all
+x ∈ A. It is a least upper bound for A if, in addition, it is the minimum of the set of all
+upper bounds of A, that is if y′ ≿ x for all x ∈ A then y′ ≿ y. If ≿ is antisymmetric, the
+least upper bound is unique and is denoted sup A. The greatest lower bound is defined
+analogously and denoted inf A.
+11We thank Dr. Hendrik S. Brandsma for providing a feedback and insightful comments.
+12Note that in decision theory, rays of a set X equipped with a reflexive and transitive binary
+relation ≿ are usually denoted with the following notation L(a, ≿) := (−∞, a] = {x ∈ X : x ≾ a} and
+U(a, ≿) := [a, +∞) = {x ∈ X : x ≿ a}, L(a, ≻) := (−∞, a) and U(a, ≻) := (a, +∞).
+19
+
+Lemma 1. Let ≿ be a total order on a connected set X. Then, X is a linear continuum
+in the order topology.13
+Proof. Suppose that a and b are two arbitrary but fixed elements of X such that a ≺ b.
+If there is no element c ∈ X such that a ≺ c ≺ b, then X is the union of the open
+rays (−∞, b) = {x ∈ X : x ≺ b} and (a, +∞) = {x ∈ X : a ≺ x} both of which are
+open sets in the order topology and are also nonempty, as the first contains a, while
+the second contains b. But this contradicts the fact that X is connected, so there must
+exists an element c ∈ X such that a ≺ c ≺ b.
+Now, to show the least upper bound property, let A be a nonempty subset of X
+such that A is bounded above in X. Let B be the set of all the upper bounds in X of
+set A, i.e.
+B := {b ∈ X : b ≿ a for every a ∈ A}
+which is nonempty. All we need to show is that B has the least element. If B has a
+smallest element (or A has a largest element, which would then be the smallest element
+of B), then that element is the least upper bound of A.
+Let us assume, instead, that B has no smallest element. Then, for any element
+b ∈ B, there exists an element b′ ∈ B such that b′ ≺ b, and so b ∈ (b′, +∞) ⊆ B with
+(b′, +∞) being an open set in X. This shows that B is a nonempty open subset of X.
+Therefore, B can be closed only in the case when B = X. But we know that B ⊂ X,
+since A ⊆ X\B and A ̸= ∅, so it cannot be the case that B = X. Therefore, B has a
+limit point b0 that does not belong to B. Then b0 is not an upper bound of set A, which
+implies the existence of an element a ∈ A such that b0 ≺ a, we can also conclude that
+b0 ∈ (−∞, a) ⊆ X\B, with (−∞, a) being an open set. This contradicts our choice of
+b0 as a limit point of set B. Therefore, the set B of all the upper bounds in X of set A
+must have a smallest element, and that element is the least upper bound of A.
+Given A ⊆ X, we denote A or ClA the topological closure of A, that is defined as
+the intersection of all closed sets containing A.
+From now on denote X as a subset of a topological space (X, τ), unless otherwise
+stated.
+Lemma 2. Let ≿ be a complete, transitive and continuous order on a connected set X.
+13Note that the converse holds as well: ≿ is a total order on a connected set X if and only if X is a
+linear continuum in the order topology.
+20
+
+Given any x, y ∈ X, with x ≻ y, we have
+x ≿ z ≿ y ⇒ z ∈ X
+for all z ∈ (X, τ)
+Proof. Suppose by contradiction that there exists z ∈ X\X such that x ≻ z ≻ y.
+By the continuity of ≿, we can partition X into two nonempty disjoint open sets
+{x ∈ X : x ≺ z} and {x ∈ X : x ≻ z}, which contradicts the connectedness of X.
+Lemma 3. Suppose that jointly ≿ and ≽ satisfy A1. If ≽ is continuous , then ≿ is
+continuous.
+Proof. For all arbitrary but fixed y, z ∈ X, by A1 we have {x ∈ X : (x, z) ≽ (y, z)} =
+{x : x ≿ y}. By A3, the set {x ∈ X : (x, z) ≽ (y, z)} is closed. Analogous is the case
+for {x : y ≿ x}, derived from A1′.
+Lemma 4. Fix y ∈ X, the set Iy := {x ∈ X : x ∼ y} is a closed set in X.
+Proof. ≿ is continuous, so for every y ∈ X we have that {x ∈ X : x ≿ y} and
+{x ∈ X : y ≿ x} are closed. Pick a point x such that x ≿ y and y ≿ x, that is x ∼ y.
+So we have {x ∈ X : x ∼ y} = {x ∈ X : x ≿ y} ∩ {x ∈ X : y ≿ x} and the intersection
+of two closed sets is closed.
+Note that when ≿ is antisymmetric, the set Iy is a singleton and Lemma 4 reduces
+to prove that X satisfies the T1 axiom of separation, that is every one-point set is closed.
+Clearly, every Hausdorff space satisfies it.
+Lemma 5. Let ≿ be a continuous total order on a connected set X. If A ⊆ X is a
+nonempty closed set in the order topology and A is bounded above (below), then supA
+(infA) belongs to A.14
+Proof. Suppose supA /∈ A. Then supA ∈ X\A, which is open. By definition, there
+exists a base element (a, b) such that
+supA ∈ (a, b) ⊆ X\A.
+A is bounded above so, by Lemma 1, sup A exists and there is an element a⋆ such that
+a ≺ a⋆ ≺ sup A, then a⋆ ∈ (a, b) ⊆ X\A, so a⋆ is an upper bound of A smaller that
+supA, reaching a contradiction. In the case X had a maximum, then consider the case
+where sup A = max X. Let U := (x, sup A] be a basic neighborhood of sup A. Then, x
+14The lemma holds even in the case we relaxed connectedness. Nevertheless, we always need to as-
+sume sup A exists. If we do not assume the existence of the least upper bound, an easy counterexample
+is N ⊂ R that is closed in the order topology, but sup N /∈ N.
+21
+
+cannot be an upper bound of A as x ≺ sup A. Hence, there exists an element a ∈ A
+such that x ≺ a ≾ sup A. Thus, as x was generic, it follows that U ∩ A ̸= ∅. This
+means that every neighborhood of sup A intersects A, that is sup A ∈ A. But A is
+closed, hence sup A ∈ A and we can conclude sup A = max A.
+The case of inf A is specular.
+Now we define the notion of convergence in any topological space.
+Definition 12. In an arbitrary topological space X, we say that a sequence x1, x2, . . .
+of points of the space X converges to the point x of X provided that, corresponding to
+each neighborhood U of x, there is a positive integer N such that xn ∈ U for all n ≥ N.
+Moreover, let ≿ a total order. We write xn ↑ x if x1 ≾ x2 ≾ · · · ≾ xn ≾ . . . and
+supnxn = x where sup is with respect to ≾. The definition xn ↓ x for a ≾-decreasing
+sequence is analogous. We say that (xn) converges monotonically to a limit point x
+when either xn ↑ x or xn ↓ x.
+We now prove one of the fundamental lemmas that allow us to generalize Shapley’s
+proof to a connected and separable subset of a topological space. Note that, as long
+as Shapley [36] is working on R, sequences as “enough” to characterize the definition
+of convergence.
+This is due to the fact that there exists a countable collection of
+neighborhoods around every point. This is not true in general, but it is for a specific
+class of spaces that are said to satisfy the first countability axiom.15 A space X is said
+to have a countable basis at the point x if there is a countable collection {Un}n∈N of
+neighborhoods of x such that any neighborhood U of x contains at least one of the sets
+Un. A space X that has a countable basis at each of its points is said to satisfy the
+first countability axiom.
+In general, however, sequences are not powerful enough to capture the idea of
+convergence we want to capture in a generic topological space. Indeed, there could
+be uncountably many neighborhoods around every point, so the countability of the
+natural number index of sequences cannot “reach” these points. The ideal solution to
+this problem is to define a more general object than a sequence, called a net, and talk
+about net-convergence. One can also define a type of object called a filter and show
+that filters also provide us a type of convergence which turns out to be equivalent to
+net-convergence. With these more powerful tools in place of sequence convergence, one
+can fully characterize the notion of convergence in any topological space.
+15There are far more general classes of spaces in which convergence can be fully characterized by se-
+quences. We refer the interested reader to the notion of Fr´echet-Urysohn spaces and Sequential spaces.
+22
+
+Nevertheless, we are now going to show that every connected, separable and totally
+ordered set X satisfies the first countability axiom. In fact, we are going to prove even
+more. We are going to show that X is metrizable, which means there exists a metric d
+on the set X that induces the topology of X.16 We give other two definitions that will
+be used to prove Lemma 6.
+Definition 13. Suppose X is T1. Then X is said to be regular (or T3) if for each pair
+consisting of a point x and a closed set B disjoint from x, there exist disjoint open sets
+containing x and B, respectively.
+Definition 14. If a space X has a countable basis for its topology, then X is said to
+satisfy the second countability axiom, or to be second-countable.
+Theorem 4 (Urysohn metrization theorem). Every regular space X with a count-
+able basis is metrizable.
+Lemma 6. Let ≿ be a continuous total order on a connected and separable topological
+space X in the order topology and A ⊆ X. We have x ∈ A if and only if there exists a
+sequence (xn) ∈ AN that converges monotonically to x.
+The steps of the proof are the following:
+(i) We show that X is regular17 and second-countable. By the Urysohn metrization
+theorem, which provides sufficient (but not necessary) conditions for a space to
+be metrizable, there exist a metric d that induces the topology of X.
+(ii) Let A ⊆ X with X metrizable, then we have that x ∈ A if and only if there exists
+a sequence of points of A converging to x.
+(iii) Finally, we use the fact that in every totally ordered topological space X, every
+sequence admits a monotone subsequence. Then, if a sequence converges, all of
+its subsequences converge to the same limit. Thus, we can extract our monotone
+converging sequence.
+Lemma 7. A totally ordered topological space X is regular in the order topology.
+Proof. It is basic topology to prove that every totally ordered set is Hausdorff, hence
+it is T1. Now, suppose x ∈ X and B is a closed set, disjoint from x. So, x ∈ X\B,
+16A metrizable space always satisfies the first countability axiom.
+17In fact, one could prove that X is also normal.
+23
+
+which is open. Then, by definition of open set, there exists a basis element (a, b) such
+that x ∈ (a, b) and (a, b) ∩ B = ∅. Pick any a0 ∈ (a, x), and let U1 = (−∞, a0) , V1 =
+(a0, ∞). If no such a0 exists (in our case it would, by connectedness of X), then let
+U1 = (−∞, x), V1 = (a, ∞). In both cases, U1 ∩ V1 = ∅. Similar is the case of the other
+side, pick b0 ∈ (x, b), and if that exists, denote U2 = (b0, ∞) , V2 = (−∞, b0) , and if
+not, let U2 = (x, ∞), V2 = (−∞, b). Again, in both cases U2 ∩ V2 = ∅. As a result, we
+obtained that, in both cases, x ∈ V1 ∩ V2 with V1 ∩ V2 open set and B ⊆ U1 ∪ U2, with
+U1 ∪ U2 open set. As V1 ∩ V2 is disjoint from U1 ∪ U2, X is regular.
+Lemma 8. A totally ordered, connected and separable topological space X is second-
+countable.
+Proof. Now we find a countable basis for the order topology of X. As X is separable,
+then let D ⊆ X be countable and dense in X, i.e. D = X. Then, define
+B := {(a, b) : a, b ∈ D with a ≺ b}
+together with, if there exists a minimal element m := min X and a maximal element
+M := max X, the set {[m, a), (a, M], a ∈ D}. In both cases, the collection B forms a
+countable base for the topology of X. To prove this, we show that for each open set
+(a, b) of the order topology of X and for every x ∈ (a, b) there is an element (a′, b′) ∈ B
+such that x ∈ (a′, b′) ⊆ (a, b).
+Suppose x ∈ (a, b) ⊂ X, then the open intervals (a, x) and (x, b) cannot be empty
+by connectedness. Hence, there exist a′ ∈ (a, x) ∩ D and b′ ∈ (x, b) ∩ D. This follows
+from the fact that D = X and x ∈ D = X if and only if every open set containing x
+intersects D. Then, it follows that x ∈ (a′, b′) ⊆ (a, b).
+Now, when m exists, suppose x = m, then x ∈ [m, a) and this set is nonempty
+by connectedness. Hence, there exists an element a′′ ∈ [m, a) ∩ D. So, it follows that
+x ∈ [m, a′′) ⊆ [m, a). Analogous is the case when M exists.
+By Lemma 7 and Lemma 8 , X satisfies all the assumptions of the Urysohn metriza-
+tion theorem, hence X is metrizable (and, a fortiori, it is first-countable).
+Lemma 9. Let A ⊆ X with X metrizable, then x ∈ A if and only if there exists a
+sequence of points of A converging to x.
+Proof. Suppose xn → x with xn ∈ A. Then, every neighborhood U of x contains a
+point of A, i.e. x ∈ A. Conversely, we use the fact that X is metrizable.18 Let x ∈ A
+18Note, once again, that here we do not need the full strength of metrizability. All we really need
+24
+
+and let d be a metric that induces the order topology. For every n ∈ N, we take the
+neighborhood Bd(x, 1
+n), of x of radius 1
+n and we choose xn to be a point such that, for
+all n, xn ∈ Bd(x, 1
+n) ∩ A. We show xn → x. Any open set U containing x contains an
+ǫ-neighborhood Bd(x, ǫ) centered at x. Choosing N such that
+1
+N < ǫ, then U contains
+xn for all n ≥ N.
+We can finally prove Lemma 6.
+Proof. The if part comes trivially by definition. If there exists a sequence that converges
+(monotonically) to x, then x ∈ A by Lemma 9.
+Conversely, if x ∈ A, then by Lemma 9 we know that there exists a sequence in A
+converging to x. Now we show that, in every totally ordered set (X, ≾), every sequence
+from N → (X, ≾) has a monotone subsequence. Indeed, this is a property that has
+nothing to do with the topology of X.
+Let (xi)i∈N be a sequence with values in X.
+We say that xk is a peak of the
+sequence if h > k ⇒ xh ≾ xk (we admit a slight abuse of notation here, as it would be
+better to call peak the index of the sequence, and not its image). We distinguish two
+cases: if there are infinitely many peaks, then the subsequence of peaks is an infinite
+non-increasing sequence and we are done. If there are only finitely many peaks, then
+let i1 be the index such that xi1 is the successor of the last peak. Then, xi1 is not a
+peak. Again, we find another index i2 > i1 such that xi2 ≿ xi1. Again, as xi2 is not a
+peak, we can find another index i3 > i2 such that xi3 ≿ xi2 ≿ xi1. Keeping defining the
+sequence in this way, we get, inductively, a non-decreasing sequence.
+In conclusion, as by assumption we have a sequence (xn) ∈ AN converging to x,
+this sequence admits a monotone subsequence. But, if a sequence converges to a point
+x, then all of its subsequences converge to the same point x. Hence, there exists a
+sequence that converges monotonically to x, proving Lemma 6.
+Note that Lemma 6 could have been proven just using the notion of first countabil-
+ity. Nevertheless, we decided to take the longer path of Urysohn metrization theorem to
+is a countable collection of neighborhoods around x. Moreover, both connectedness and separability
+are not necessary conditions. We refer the interested reader to the nice two-page paper of Lutzer [25],
+that proves a linearly ordered space X is metrizable in the order topology if and only if the diagonal
+∆ := {(x, x) : x ∈ X} is a countable intersection of open subsets of X × X, i.e. the diagonal is a Gδ
+set. Furthermore, this condition can be shown to be equivalent to have a σ-locally countable basis,
+which is a condition more in the spirit of the Nagata-Smirnov metrization theorem which requires a
+σ-locally finite basis.
+25
+
+show how “well-behaved” a totally ordered, connected and separable topological space
+can be.
+Lemma 10. Let (X, ≿) be a topological space with the order topology. Let ≽ be another
+order relation on X × X such that A1 and A3 hold,19 and suppose (xn), (yn) converge
+to x and y respectively, and (wn), (zn) converge to w and z respectively. If for every
+n ∈ N we have (xn, yn) ≽ (wn, zn) then (x, y) ≽ (w, z).
+Proof. Denote the set A := {(x, y, w, z) ∈ X × X × X × X : (x, y) ≽ (w, z)} and pick
+a sequence of points with values in A, that is pick (xn, yn, wn, zn) ∈ AN converging to
+(x, y, w, z). By assumption, we have that xn → x, yn → y, wn → w, zn → z and this
+is equivalent to (xn, yn, wn, zn) → (x, y, w, z). Indeed, a sequence in the product space
+X × X × X × X converges to (x, y, w, z) if and only if it converges componentwise, i.e.
+xn → x, yn → y, wn → w, zn → z. We now prove this fact.
+Assume (xn, yn, wn, zn) → (x, y, w, z) in X ×X ×X ×X. Let U1, U2, U3, U4 be open
+sets containing x, y, w, z, respectively. Then U1×U2×U3 ×U4 is a basis element (hence,
+open) for the product topology containing (x, y, w, z). By definition of convergence, we
+can find n0 such that for all n ≥ n0 we have (xn, yn, wn, zn) ∈ U1 × U2 × U3 × U4.
+Thanks to the fact that projections are continuous functions, they preserve convergent
+sequences and so for all n ≥ n0 we have xn ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ U4, i.e.
+xn → x, yn → y, wn → w, zn → z.
+Conversely, if xn → x, yn → y, wn → w, zn → z, let U⋆ be an open subset of
+X×X×X×X such that (x, y, w, z) ∈ U⋆. By definition of product topology, we can find
+U1 ⊆ X open in X, . . . , U4 ⊆ X open in X such that x ∈ U1, y ∈ U2, w ∈ U3, z ∈ U4. By
+convergence, we have that for all i = 1, 2, 3, 4 there exists nki ∈ N such that for all n ≥
+nki we have xn ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ U4. Now pick N := max{nk1, nk2, nk3, nk4}
+and for every n ≥ N we have (xn, yn, wn, zn) ∈ U1 × U2 × U3 × U4 ⊆ U⋆. Hence, by
+definition of convergence, (xn, yn, wn, zn) → (x, y, w, z).
+Now we want to show (x, y, w, z) ∈ A, with A closed in the product topology.
+We now prove that every closed set in the product topology is sequentially closed.20
+This means we want to show that if we pick a sequence of points (xn, yn, wn, zn) with
+values in A ⊆ X that is converging to a point (x, y, w, z) ∈ X, then (x, y, w, z) ∈ A.
+Pick a sequence (xn, yn, wn, zn) with values in A ⊆ X that is converging to a point
+19Note that the order topology and A1 are redundant assumptions. The lemma follows immediately
+by continuity of ≽ alone.
+20Note that when X is metrizable, a set C ⊆ X is closed ⇐⇒ C is sequentially closed.
+26
+
+(x, y, w, z) ∈ X. Then, let U⋆ be any neighborhood of (x, y, w, z). By convergence,
+there exist an n0 ∈ N such that for all n ≥ n0 we have (xn, yn, wn, zn) ∈ U⋆ and, in
+particular, (xn, yn, wn, zn) ∈ U⋆ ∩ A. Since U⋆ was an arbitrary but fixed neighborhood
+of (x, y, w, z), then (x, y, w, z) is in the closure of A, i.e. (x, y, w, z) ∈ A. But A is
+closed, therefore A = A, so (x, y, w, z) ∈ A, hence (x, y) ≽ (w, z).
+The proof of Theorem 5 in chapter 3, as in the original version of Shapley [36],
+relies on two very interesting lemmas. Similar propositions have been taken as axioms
+in environments that lack the topological assumptions on the set of alternatives X.
+Lemma 11. Let (w,z) be an element of X × X. If x′, x′′, y ∈ X are such that:
+(x′, y) ≽ (w, z) ≽ (x′′, y)
+(7)
+then there exists a unique, up to indifference, x⋆ ∈ X such that
+(x⋆, y) ∼ (w, z)
+(8)
+and x′ ≿ x⋆ ≿ x′′.
+Proof. Define x0 := inf{x ∈ X : (x, y) ≽ (w, z)} and denote A := {x ∈ X : (x, y) ≽
+(w, z)} this set. The set A is nonempty as x′ ∈ A, A is bounded below by x′′ as we
+have (w, z) ≽ (x′′, y) and, by transitivity and A1, we reach x ≿ x′′ for every x ∈ A.
+Thus, x0 is such that x′ ≿ x0 ≿ x′′ and so x0 ∈ X by Lemma 2. Analogously, we define
+x0 := sup{x ∈ X : (w, z) ≽ (x, y)} and denote B := {x ∈ X : (w, z) ≽ (x, y)} this set.
+Then, B is nonempty as x′′ ∈ B, B is bounded above by x′ as we have (x′, y) ≽ (w, z)
+and, by transitivity and A1, we reach x′ ≿ x for every x ∈ B. Thus, x0 is such that
+x′ ≿ x0 ≿ x′′ and so x0 ∈ X by Lemma 2.
+By A3, the sets A and B are closed and so, by Lemma 5, we have x0 ∈ A and
+x0 ∈ B so that
+(x0, y) ≽ (w, z) ≽ (x0, y)
+By transitivity and by A1 we have x0 ≿ x0.
+Assume now by contradiction that x0 ≻ x0. By Lemma 1 there exists x⋆ ∈ X such
+that x0 ≺ x⋆ ≺ x0. But then, comparing x⋆ with (w, z), (x⋆, y) ≽ (w, z) can hold only
+if x0 ∼ x⋆ ≻ x0, so x0 ∼ x⋆ and therefore x⋆ should be the infimum of A, reaching a
+contradiction. Specular is the contradiction in the other case. Therefore, as there does
+not exist any x⋆ ∈ X such that x0 ≺ x⋆ ≺ x0, we must conclude that x0 ∼ x0. By
+transitivity and A1 we have
+(x0, y) ∼ (w, z) ∼ (x0, y)
+27
+
+This proves the existence of x⋆ ∈ X for which (8) holds.
+Let x ∈ X be any other element of X for which (8) holds. By transitivity, (x⋆, y) ∼
+(x, y). By A1, we have x⋆ ∼ x and this completes the proof.
+Lemma 12. Let x, z ∈ X such that x ≻ z. Then, there exists a unique, up to indiffer-
+ence, y⋆ ∈ X such that
+(x, y⋆) ∼ (y⋆, z)
+and x ≻ y⋆ ≻ z.
+Proof. Define y0 to be the least upper bound of the set C := {y ∈ X : (x, y) ≽ (y, z)}.
+This set is nonempty as if we pick y = z we have (x, z) ≽ (z, z) that by A1 is equivalent
+to x ≿ z, that holds by assumption. C is also bounded from above by x as if we pick
+y = x we have (x, x) ≽ (x, z) that by A1′ is equivalent to z ≿ x, that, by completeness,
+contradicts the assumption of x ≻ z showing that x is an upper bound for C. Since C
+is nonempty and bounded above by x, by Lemma 2 we have y0 ∈ X.
+Similarly, by defining y0 to be the greatest lower bound of the set D := {y ∈ X :
+(y, z) ≽ (x, y)}. This set is nonempty as if we pick y = x we have (x, z) ≽ (x, x) that
+by A1′ is if and only if x ≿ z, that holds by assumption. This set is also bounded
+from below by z as if we pick y = z we have (z, z) ≽ (x, z) that by A1 is if and only
+if z ≿ x, that, by completeness, contradicts the assumption of x ≻ z showing that z is
+a lower bound for D. Since D is nonempty and bounded below by z, by Lemma 2 we
+have y0 ∈ X.
+By A3 the sets C and D are closed, so by Lemma 5 we have y0 ∈ C and y0 ∈ D,
+that is
+(x, y0) ≽ (y0, z) and (y0, z) ≽ (x, y0)
+(9)
+We show now that y0 ≿ y0. Suppose, by contradiction, y0 ≻ y0. By Lemma 1 there
+exists y⋆ ∈ X such that y0 ≻ y⋆ ≻ y0. Then, by definition of y0 we have (y⋆, z) ≺ (x, y⋆),
+while by the definition of y0 we have (x, y⋆) ≺ (y⋆, z). This contradiction shows that
+y0 ≿ y0. By A1 this is equivalent to
+(y0, z) ≽ (y0, z) for all z ∈ X.
+(10)
+By A1′ it is also equivalent to
+(x, y0) ≽ (x, y0) for all x ∈ X.
+(11)
+Putting together equation 9 with equations 10 and 11, we reach the loop
+(y0, z) ≽ (y0, z) ≽ (x, y0) ≽ (x, y0) ≽ (y0, z).
+28
+
+By transitivity, we have (y0, z) ∼ (y0, z) and (x, y0) ∼ (x, y0). By A1, we conclude that
+y0 ∼ y0.
+We conclude proving that from A1, A2 and A3 we can derive A1′.
+Lemma 13. Let X be a connected subset of a topological space.
+If ≿ is complete
+and transitive, ≽ is complete, transitive, satisfies A3 and A2, and jointly ≿ and ≽
+satisfy A1, then A1′ holds, that is, for all x, y, z ∈ X we have x ≿ y if and only if
+(z, y) ≽ (z, x).
+Proof. By contradiction, suppose A1′ fails. Then, there exist x, y, z ∈ X such that
+(z, y) ≽ (z, x) and x ≺ y. We consider two cases: y ≻ z and y ≾ z.
+If y ≻ z then, being (z, y) ≽ (z, x) by assumption, we have
+(x, x) ∼ (y, y) ≻ (z, y) ≽ (z, x)
+by A2 and A1, respectively. We apply Lemma 11 to find a w ∈ X such that
+(w, x) ∼ (z, y) and x ≿ w ≿ z.
+Being y ≻ x, we have
+(z, z) ∼ (y, y) ≻ (x, y) ∼ (w, z) ≿ (z, z)
+by A2, A1, A2, A1, respectively. This implies a contradiction in the case y ≻ z.
+Assume now y ≾ z. Being y ≾ z and x ≺ y, by transitivity we have x ≺ z. We
+can proceed as in the previous case, interchanging the roles of x and y and reversing
+all the inequalities.
+3
+The theorem
+We can now state and prove Shapley’s theorem in our general version.
+Theorem 5. Let X be a connected and separable subset of a topological space. If ≿ is
+complete and transitive, ≽ is complete, transitive, satisfies A2 and A3, and jointly ≿
+and ≽ satisfy A1, then the pair (≿, ≽) can be represented by a continuous measurable
+utility function u: X → R, that is, for each pair x, y ∈ X,
+x ≿ y ⇐⇒ u(x) ≥ u(y)
+(12)
+and for each quadruple x, y, z, w ∈ X,
+(x, y) ≽ (z, w) ⇐⇒ u(x) − u(y) ≥ u(z) − u(w).
+(13)
+Moreover, u is unique up to positive affine transformations.
+29
+
+Proof. We first prove the result when ≿ is antisymmetric. In view of Lemma 1, through-
+out the proof we will consider suprema and infima of subsets of X.
+Suppose X is not a singleton, otherwise the result is trivially true. Let a0, a1 ∈ X
+be two distinct elements of X such that, without loss of generality, a1 ≻ a0.
+Assign u(a0) = 0 and u(a1) = 1. Now we want to show that u has a unique
+extension on X which is a measurable utility function for (≿, ≽). To ease notation,
+denote
+1 := (a1, a0) , 0 := (a0, a0) , −1 := (a0, a1).
+Clearly, 1, 0, −1 ∈ X × X and, by A1 and A1′, 1 ≻ 0 ≻ −1. Then, by A2 we have
+(x, x) ∼ 0 for every x ∈ X. Moreover, for every y ∈ X we have either:
+(i) There exists a unique T1(y) ∈ X such that (T1(y), y) ∼ 1
+or
+(ii) 1 ≻ (x, y) for all x ∈ X
+Indeed, if (ii) fails, there exists x′ ∈ X such that (x′, y) ≽ 1. Since (x′, y) ≽ 1 ≽ 0 ∼
+(y, y), by Lemma 11 there exists an element T1(y) ∈ X such that (T1(y), y) ∼ 1. By
+A1 and antisymmetry of ≿ , (T1(y), y) ∼ (y′, y) implies T1(y) = y′, so T1(y) is unique.
+In addition, note that y ≺ T1(y). Indeed, (y, y) ∼ 0 ≺ 1 ∼ (T1(y), y), and so A1
+implies y ≺ T1(y). In a similar way as before, for every y ∈ X we have either:
+(i.bis) There exists a unique T−1(y) ∈ X such that (T−1(y), y) ∼ −1
+or
+(ii.bis) −1 ≺ (x, y) for all x ∈ X
+Indeed, if (ii.bis) fails, there exists x′ ∈ X such that (x′, y) ≼ −1. Since (x′, y) ≼ −1 ≼
+0 ∼ (y, y), by Lemma 11 there exists an element T−1(y) ∈ X such that (T1(y), y) ∼ −1.
+By A1 and antisymmetry of ≿ , (T−1(y), y) ∼ (y′, y) implies T−1(y) = y′, so T−1(y) is
+unique.
+In addition, note that T−1(y) ≺ y. Indeed, (T−1(y), y) ∼ −1 ≺ 0 ∼ (y, y), and so
+A1 implies T−1(y) ≺ y.
+Now define a2 := T1(a1) if (i) holds for y = a1, i.e. if there exists a unique
+T1(a1) ∈ X such that (T1(a1), a1) ∼ 1. Similarly, set a3 := T1(a2) if (i) holds for y = a2,
+and continue in this way till (if ever) occurs y = an for which (ii) holds, i.e. 1 ≻ (x, an)
+for every x ∈ X. Analogously, we define a−1 := T−1(a0) if (i.bis) holds for y = a0, set
+a−2 := T−1(a−1) if (i.bis) holds for y = a−1, and continue in this way till (if ever) occurs
+y = a−n for which (ii.bis) holds.
+30
+
+Now define A := {. . . , a−2, a−1, a0, a1, a2, . . . }, with
+· · · ≺ a−2 ≺ a−1 ≺ a0 ≺ a1 ≺ a2 ≺ . . .
+The set A can be finite or infinite in either direction. If we consider now a sequence that
+from an index set Pa ⊆ Z maps to A, we define the following function a : Pa ⊆ Z → A.
+Now we start to extend u to A. Define the following:
+u(ap) = p
+for every p ∈ Pa.
+Clearly, we have (12), i.e. x ≿ y if and only if u(x) ≥ u(y) for every x, y that are images
+of the sequence a, so (12) holds on A.
+Now we show that (13) holds whenever x, y, z, w ∈ A ⊂ X, say x = ap, y = aq, z =
+ap−d where p, q, p − d ∈ Pa. Without loss of generality, assume d ≥ 0. We first prove
+the “equality” case of (13), that is
+(x, y) ∼ (z, w) ⇐⇒ u(x) − u(y) = u(z) − u(w)
+(14)
+By construction we have
+(ap, ap−1) ∼ 1 ∼ (aq, aq−1)
+so, by transitivity and A2, we have:
+(ap, aq) ∼ (ap−1, aq−1)
+Iterating this procedure finitely many times we reach:
+(x, y) = (ap, aq) ∼ (ap−d, aq−d) = (z, aq−d)
+(15)
+By transitivity, (z, aq−d) ∼ (z, w) and so, by A1 aq−d = w, so that u(aq−d) = u(w).
+By definition of u we can write
+u(x) − u(y) = u(ap) − u(aq) = p − q = u(ap−d) − u(aq−d) = u(z) − u(w)
+thus proving (14). Next we prove
+(x, y) ≻ (z, w) ⇐⇒ u(x) − u(y) > u(z) − u(w)
+(16)
+By transitivity, (z, aq−d) ≻ (z, w) and so, by A1′, w ≻ aq−d, so that u(w) > u(aq−d).
+By definition of u, from (15) we can write
+u(x) − u(y) = u(ap) − u(aq) = p − q = u(ap−d) − u(aq−d) > u(z) − u(w)
+thus proving (16).
+Summing up, both (12) and (13) hold on the terms of the set A. Using Lemma
+12, now we want to extend u to the points of X that lie between terms of the set A.
+Set b0 := a0 and since a1 ≻ a0, by Lemma 12 there exists b1 ∈ X, with a1 ≻ b1 ≻ a0,
+31
+
+such that
+(a1, b1) ∼ (b1, a0)
+Now build the set B := {. . . , b−2, b−1, b0, b1, b2, . . . }, with
+· · · ≺ b−2 ≺ b−1 ≺ b0 ≺ b1 ≺ b2 ≺ . . .
+based on b0, b1, in the same way we constructed A from a0, a1. Also here, we can define
+a sequence that from an index set Pb ⊆ Z maps to B, that is, we define the following
+function b : Pb ⊆ Z → B.
+By construction we have
+(b2, b1) ∼ (b1, b0)
+Together with (a1, b1) ∼ (b1, a0), by transitivity we have (b2, b1) ∼ (a1, b1). By A1,
+b2 = a1. Analogously, one can verify that
+b2p = ap for every p ∈ Pa
+(17)
+So, the terms of the set B lie between the terms of the set A, i.e. the set B refines
+A and we can write
+A ⊆ B
+(18)
+Denote now c0 := b0 = a0 and we let c1 ∈ X be that element provided by Lemma
+12 such that (b1, c1) ∼ (c1, b0). In the same way we constructed B from A, we can
+construct, from B, a third set C := {. . . , c−2, c−1, c0, c1, c2, . . . }, with
+· · · ≺ c−2 ≺ c−1 ≺ c0 ≺ c1 ≺ c2 ≺ . . .
+based on c0, c1. We can see that
+c2p = bp for every p ∈ Pc
+where Pc ⊆ Z is the collection of indexes of the sequence c : Pc ⊆ Z → C.
+The set C refines B
+B ⊆ C
+(19)
+We keep iterating this process, constructing sets that refine one another and, for
+ease of notation, we denote them in the following way:
+A0 := A
+and
+a0
+p := ap ∈ A0
+A1 := B
+and
+a1
+p := bp ∈ A1
+A2 := C
+and
+a2
+p := cp ∈ A2
+· · ·
+32
+
+These sets generalize the inclusions (18) and (19) as follows:
+A0 ⊆ A1 ⊆ A2 ⊆ · · · ⊆ An ⊆ . . .
+(20)
+So, in general, an
+p for p ̸= 1 is obtained from the construction of (i) and (ii), applied to
+the points a0, an
+1. The term an
+1, for n > 0, is the “midpoint” between an−1
+1
+and a0, that
+exists by Lemma 12. By iterating the construction of (17), we have that
+p
+2n = q
+2m =⇒ an
+p = am
+q
+In the spirit of (20), we extend u to all points in A∞ := �∞
+n=1 An by:
+u(an
+p) = p
+2n
+for all an
+p ∈ An
+Relations (12) and (13) hold in this extended domain: given x, y, z, w ∈ �∞
+n=1 An,
+just take n large enough so that they become, up to indifference, terms of the set An
+and proceed in the same exact way as we did for the set A0.
+To complete the construction of u we only remain to show A∞ is dense in X,
+that is A∞ = X. We first show that none of the sets An has, for its sequences of
+points an, a point of accumulation in X. Indeed, fix n and suppose by contradiction
+that an
+pk converges monotonically to a⋆ ∈ X, where, without loss of generality, we
+assume an
+pk ↑ a⋆ with a⋆ ∈ X, i.e. (pk) is an increasing sequence of integers. Denote
+1n := (an
+1, a0) and we have, for every k ∈ N,
+(an
+1+pk, an
+pk) ≽ 1n ≻ 0
+By Lemma 10, we have (a⋆, a⋆) ≽ 1n. So, by transitivity, we reach (a⋆, a⋆) ≻ 0, a
+contradiction. We conclude that, fixed n, none of the sequences an with values in An
+has a limit point in X.
+To prove A∞ = X, the implication A∞ ⊆ X is trivial by construction. Now we
+want to show A∞ ⊇ X, that is all the elements of X belong to the closure of A∞
+as well. Fix x ∈ X such that, without loss of generality, x ≿ a0. For n ≥ 1, define
+yn := sup{y ∈ An : x ≿ y}. Note that a0 ∈ {y ∈ An : x ≿ y}, so this set is nonempty
+and we can write x ≿ yn ≿ a0. By Lemma 2, yn ∈ X. Note further that, as shown
+before, An cannot have accumulation points in X so, as long as yn ∈ X, it follows
+yn cannot be an accumulation point of An. So, yn must belong to An and we denote
+yn := an
+pn. As a result, we have:
+an
+pn−k ≾ x ≺ an
+pn+k for every k > 0
+(21)
+We also have that
+1n ≻ (x, yn)
+(22)
+33
+
+Indeed, if (22) were not true, then (x, an
+pn) ≽ 1n. We consider two cases: an
+1+pn ≻ x or
+an
+1+pn ≾ x. If an
+1+pn ≻ x, then, thanks to A1, we reach the following contradiction:
+1n ∼ (an
+1+pn, an
+pn) ≻ (x, an
+pn) ≽ 1n
+(23)
+So an
+1+pn ≾ x, but this contradicts (21), that is, it contradicts yn to be the supremum.
+Thus, (22) holds. In particular, by A1 and A2, we can write (x, yn) ≽ (yn, yn) ∼ 0,
+leading to
+1n ≻ (x, yn) ≽ 0
+(24)
+Now, when n → ∞, as the sets An+1 ⊇ An ⊇ An−1 . . . are nested one into the
+other by (20), we can write, for every n ≥ 1, yn ≾ yn+1 ≾ x. Thus, the points yn form
+a non-decreasing sequence that is bounded from above by x. Call y⋆ the limit of this
+sequence, that is well-defined by Lemma 1. Since a0 ≾ y⋆ ≾ x, by Lemma 2 it follows
+that y⋆ ∈ X. In particular, by Lemma 6 we have y⋆ ∈ A∞, because, for every fixed
+n ≥ 1, yn is a term of the sets An, and so (yn) ∈ AN
+∞.
+As to the 1n terms, for n fixed, we see that
+1n ∼ (an
+2, an
+1) ∼ (an−1
+1
+, an
+1)
+We also have that, for every n ≥ 1, a0 ≾ an+1
+1
+≾ an
+1.
+Thus, the points an
+1 form, for n → ∞, a non-increasing sequence that is bounded
+from below by a0. Call a⋆ the limit of this sequence, that is well-defined by Lemma 1.
+Since a0 ≾ a⋆ ≾ a1, by Lemma 2 we have a⋆ ∈ X.
+Consider now (an−1
+1
+, an
+1) and (x, yn). By Lemma 10 and from (24) it follows that
+(a⋆, a⋆) ≽ (x, y⋆) ≽ 0
+Since, by A2, (a⋆, a⋆) ∼ 0, by transitivity (x, y⋆) ∼ 0, so that x ∼ y⋆, i.e. x = y⋆ as ≿
+is antisymmetric.
+Since x was arbitrarily chosen in X and y⋆ ∈ A∞, we can conclude x ∈ A∞, so
+that A∞ = X. Therefore, we can extend u by continuity to the whole set X by setting
+u(x) = lim
+n→∞ u(xn)
+if (xn) ∈ AN
+∞ converges monotonically to x. Note that u : X → R is well-defined.
+Indeed, to prove it is well-posed we show that if xn and yn are two sequences that
+converge to x, then limn→∞ u(xn) = limn→∞ u(yn). This follows easily by continuity of
+u.21 In light of Lemma 6, it is easy to see that u satisfies (12) and (13).
+As to uniqueness, observe that any other u that satisfies (12) and (13) can be
+21Recall that in every topological space X continuity implies sequential continuity. The converse
+holds if X is first-countable.
+34
+
+normalized so that u(a0) = 0 and u(a1) = 1. So, u must agree on u at each step of
+the constructive procedure for u just seen. Indeed, for a given u : X → R, define the
+following positive affine transformation f : Im(u) → R such that
+f(x) :=
+x − u(a0)
+u(a1) − u(a0)
+It is immediate to see that, for the equivalent utility function �u := f ◦ u, we have
+�u(a0) = 0 and �u(a1) = 1.
+Summing up, we proved Theorem 5 if ≿ is antisymmetric.
+Now we drop this
+assumption. Let X/∼ be the quotient space with respect to the equivalence relation
+∼. The set {x ∈ X : x ∼ y} is a closed set in X by Lemma 4, so (X/∼, ˜≿) is a totally
+ordered connected and separable subset of a topological space, where ˜≿ is the total
+order induced by the weak order ≿.22 Therefore, the orders ≿ and ≽ induce orders ˜≿
+and ˜≽ on the quotient set X/∼, by setting, for all [x], [y] ∈ X/∼
+[x] ˜≿ [y] ⇐⇒ x ≿ y
+and, for all [x], [y], [z], [w] ∈ X/∼
+([x], [y]) ˜≽ ([z], [w]) ⇐⇒ (x, y) ≽ (z, w)
+It is routine to show that the orders ˜≿ over X/∼ and ˜≽ over X/∼ × X/∼ inherit the
+same properties of ≿ and ≽ used in the theorem. So, by what has been proved so far,
+there exists ˜u : X/∼ → R that satisfies (12) and (13) for (˜≿, ˜≽). Let π : X → X/∼ be
+the quotient map. Then, the function u : X → R defined as u = ˜u ◦ π is a well-defined
+measurable utility function, i.e. it is easily seen to satisfy (12) and (13) for (≿, ≽).
+To conclude, we show that u satisfies (12) and (13). If x ∼ y then [x] = [y] and,
+by the theorem we have just proved, ˜u([x]) = ˜u([y]), which is (˜u ◦ π)(x) = (˜u ◦ π)(y),
+and so u(x) = u(y). If x ≻ y, then [x] ≻ [y], which implies ˜u([x]) > ˜u([y]), which is
+(˜u ◦ π)(x) > (˜u ◦ π)(y), and so u(x) > u(y).
+Conversely, assume u(x) ≥ u(y) and suppose by contradiction x � y that, by
+completeness, is y ≻ x. If u(x) = u(y) then ˜u([x]) = ˜u([y]) ⇐⇒ [x] = [y] ⇐⇒ x ∼ y,
+a contradiction. If u(x) > u(y) then ˜u([x]) > ˜u([y]) ⇐⇒ [x] > [y] ⇐⇒ x ≻ y, a
+contradiction. Hence, (12) holds for u.
+By definition, we have that ([x], [y]) ≽ ([z], [w])
+⇐⇒
+(x, y) ≽ (z, w), for all
+[x],[y],[z],[w] ∈ X/∼. So, we can write (x, y) ≽ (z, w) ⇐⇒ ([x], [y]) ≽ ([z], [w]) ⇐⇒
+˜u([x])−˜u([y]) ≥ ˜u([z])−˜u([w]) ⇐⇒ u(x)−u(y) ≥ u(z)−u(w). Hence, also (13) holds
+for u.
+22That is, ˜≿ := ≿ /∼ ⊆ X/∼ × X/∼.
+35
+
+This completes the proof of Theorem 5.
+Graphically, we can build the following diagram to represent our construction.
+X
+X/∼
+R
+π
+u
+˜u
+References
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+39
+

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+arXiv:2301.03137v1  [math.NT]  9 Jan 2023
+Gaps on the intersection numbers of
+sections on a rational elliptic surface
+Renato Dias Costa
+Abstract
+Given a rational elliptic surface X over an algebraically closed field, we investigate whether a
+given natural number k can be the intersection number of two sections of X. If not, we say that
+k a gap number. We try to answer when gap numbers exist, how they are distributed and how to
+identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation
+to the classical problem of representing integers by positive-definite quadratic forms.
+Contents
+1
+Introduction
+2
+2
+Preliminaries
+4
+2.1
+The Mordell-Weil Lattice
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Gap numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.3
+Bounds cmax, cmin for the contribution term . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.4
+The difference ∆ = cmax − cmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+2.5
+The quadratic form QX
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+3
+Intersection with a torsion section
+10
+4
+Existence of a pair of sections with a given intersection number
+11
+4.1
+Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+4.2
+Sufficient conditions when ∆ ≤ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.2.1
+The case ∆ < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.2.2
+The case ∆ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+4.3
+Necessary and sufficient conditions for ∆ ≤ 2
+. . . . . . . . . . . . . . . . . . . . . .
+14
+4.4
+Summary of sufficient conditions
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+5
+Main Results
+15
+5.1
+No gap numbers in rank r ≥ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+5.2
+Gaps with probability 1 in rank r ≤ 2
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+5.3
+Identification of gaps when E(K) is torsion-free with rank r = 1
+. . . . . . . . . . .
+18
+5.4
+Surfaces with a 1-gap
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+6
+Appendix
+23
+1
+
+1
+Introduction
+Description of the problem. Let X be a rational elliptic surface over an algebraically closed
+field, i.e. a smooth, rational projective surface with a fibration π : X → P1 whose general fiber
+is a smooth curve of genus 1. Assume also that π is relatively minimal, i.e. no fiber contains an
+exceptional curve in its support. We use E/K to denote the generic fiber of π, which is an elliptic
+curve over the function field K := k(P1). By the Mordell-Weil theorem, the set E(K) of K-points
+is a finitely generated Abelian group, whose rank we denote by r. The points on E(K) are in
+bijective correspondence with the sections of π, as well as with the exceptional curves on X, so
+we use these terms interchangeably. This paper addresses the following question: given sections
+P1, P2 ∈ E(K), what values can the intersection number P1 · P2 possibly attain?
+Original motivation.
+The problem originates from a previous investigation of conic bundles
+on X, i.e. morphisms ϕ : X → P1 whose general fiber is a smooth curve of genus zero [Cos]. More
+specifically, one of the ways to produce a conic bundle is by finding a pair of sections P1, P2 ∈ E(K)
+with P1 · P2 = 1, so that the linear system |P1 + P2| induces a conic bundle ϕ|P1+P2| : X → P1
+having P1 + P2 as a reducible fiber. We may ask under which conditions such a pair exists. An
+immediate necessary condition is that r ≥ 1, for if r = 0 any two distinct sections must be disjoint
+[SS19, Cor. 8.30]. Conversely, given that r ≥ 1, does X admit such a pair? The first observation
+is that r ≥ 1 implies an infinite number of sections, so we should expect infinitely many values for
+P1·P2 as P1, P2 run through E(K). Then the question is ultimately: what values may P1·P2 assume?
+Mordell-Weil lattices. The computation of intersection numbers on a surface is a difficult prob-
+lem in general. However, as we are concerned with sections on an elliptic surface, the information
+we need is considerably more accessible. The reason for this lies in the Mordell-Weil lattice, a
+concept first established in [Elk90], [Shi89], [Shi90]. It involves the definition of a Q-valued pair-
+ing ⟨·, ·⟩ on E(K), called the height pairing [SS19, Section 6.5], inducing a positive-definite lattice
+(E(K)/E(K)tor, ⟨·, ·⟩), named the Mordell-Weil lattice.
+A key aspect of its construction is the
+connection with the Néron-Severi lattice, so that the height pairing and the intersection pairing
+of sections are strongly intertwined. In the case of rational elliptic surfaces, the possibilities for
+the Mordell-Weil lattice have already been classified in [OS91], which gives us a good starting point.
+Representation of integers.
+The use of Mordell-Weil lattices in our investigation leads to
+a classical problem in number theory, which is the representation of integers by positive-definite
+quadratic forms. Indeed, the free part of E(K) is generated by r terms, so the height h(P) := ⟨P, P⟩
+induces a positive-definite quadratic form on r variables with coefficients in Q. If O ∈ E(K) is the
+neutral section and R is the set of reducible fibers of π, then by the height formula (2)
+h(P) = 2 + 2(P · O) −
+�
+v∈R
+contrv(P),
+where the sum over v is a rational number which can be estimated. By clearing denominators,
+we see that the possible values of P · O depend on a certain range of integers represented by a
+positive-definite quadratic form with coefficients in Z. This point of view is explored in some parts
+of the paper, where we apply results such as the classical Lagrange’s four-square theorem [HW79,
+§20.5], the counting of integers represented by a binary quadratic form [Ber12, p. 91] and the more
+recent Bhargava-Hanke’s 290-theorem on universal quadratic forms [BH, Thm. 1].
+2
+
+Statement of results. Given k ∈ Z≥0 we investigate whether there is a pair of sections P1, P2 ∈
+E(K) such that P1 · P2 = k. If such a pair does not exist, we say that X has a k-gap, or that k is
+a gap number. Our first result is a complete identification of gap numbers in some cases:
+Theorem 5.7. If E(K) is torsion-free with rank r = 1, we have the following characterization of
+gap numbers on X according to the lattice T associated to the reducible fibers of π.
+T
+k is a gap number ⇔ none of
+the following are perfect squares
+E7
+k + 1, 4k + 1
+A7
+k+1
+4 , 16k, ..., 16k + 9
+D7
+k+1
+2 , 8k + 1, ..., 8k + 4
+A6 ⊕ A1
+k+1
+7 , 28k − 3, ..., 28k + 21
+E6 ⊕ A1
+k+1
+3 , 12k + 1, ..., 12k + 9
+D5 ⊕ A2
+k+1
+6 , 24k + 1, ..., 24k + 16
+A4 ⊕ A3
+k+1
+10 , 40k − 4, ..., 40k + 25
+A4 ⊕ A2 ⊕ A1
+k+1
+15 , 60k − 11, ..., 60k + 45
+We also explore the possibility of X having no gap numbers. We prove that, in fact, this is
+always the case if the Mordell-Weil rank is big enough.
+Theorem 5.2. If r ≥ 5, then X has no gap numbers.
+On the other hand, for r ≤ 2 we show that gap numbers occur with probability 1.
+Theorem 5.4. If r ≤ 2, then the set of gap numbers of X, i.e. G := {k ∈ N | k is a gap number of X}
+has density 1 in N, i.e.
+lim
+n→∞
+#G ∩ {1, ..., n}
+n
+= 1.
+At last we answer the question from the original motivation, which consists in classifying the
+rational elliptic surfaces with a 1-gap:
+Theorem 5.8. X has a 1-gap if and only if r = 0 or r = 1 and π has a III∗ fiber.
+3
+
+Structure of the paper. The text is organized as follows. Section 2 introduces the main objects,
+namely the Mordell-Weil lattice, the bounds cmax, cmin for the contribution term, the difference
+∆ = cmax −cmin and the quadratic form QX induced by the height pairing. In Section 3 we explain
+the role of torsion sections in the investigation. The key technical results are gathered in Section 4,
+where we state necessary and sufficient conditions for having P1 · P2 = k for a given k. Section 5
+contains the main results of the paper, namely: the description of gap numbers when E(K) is
+torsion-free with r = 1 (Subsection 5.3), the absence of gap numbers for r ≥ 5 (Subsection 5.1),
+density of gap numbers when r ≤ 2 (Subsection 5.2) and the classification of surfaces with a 1-gap
+(Subsection 5.4). Section 6 is an appendix containing Table 8, which stores the relevant information
+about the Mordell-Weil lattices of rational elliptic surfaces with r ≥ 1.
+2
+Preliminaries
+Throughout the paper X denotes a rational elliptic surface over an algebraically closed field
+k of any characteristic. More precisely, X is a smooth rational projective surface with a fibration
+π : X → P1, with a section, whose general fiber is a smooth curve of genus 1. We assume moreover
+that π is relatively minimal (i.e. each fiber has no exceptional curve in its support) [SS19, Def.
+5.2]. The generic fiber of π is an elliptic curve E/K over K := k(P1). The set E(K) of K-points is
+called the Mordell-Weil group of X, whose rank is called the Mordell-Weil rank of X, denoted by
+r := rank E(K).
+In what follows we introduce the main objects of our investigation and stablish some notation.
+2.1
+The Mordell-Weil Lattice
+We give a brief description of the Mordell-Weil lattice, which is the central tool used in the
+paper. Although it can be defined on elliptic surfaces in general, we restrict ourselves to rational
+elliptic surfaces. For more information on Mordell-Weil lattices, we refer the reader to the com-
+prehensive introduction by Schuett and Shioda [SS19] in addition to the original sources, namely
+[Elk90], [Shi89], [Shi90].
+We begin by noting that points in E(K) can be regarded as curves on X and by defining the
+lattice T and the trivial lattice Triv(X), which are needed to define the Mordell-Weil lattice.
+Sections, points on E(K) and exceptional curves. The sections of π are in bijective cor-
+respondence with points on E(K). Moreover, since X is rational and relatively minimal, points on
+E(K) also correspond to exceptional curves on X [SS10, Section 8.2]. For this reason we identify
+sections of π, points on E(K) and exceptional curves on X.
+The lattice T and the trivial lattice Triv(X). Let O ∈ E(K) be the neutral section and
+R := {v ∈ P1 | π−1(v) is reducible} the set of reducible fibers of π. The components of a fiber
+π−1(v) are denoted by Θv,i, where Θv,0 is the only component intersected by O. The Néron-Severi
+group NS(X) together with the intersection pairing is called the Néron-Severi lattice.
+4
+
+We define the following sublattices of NS(X), which encode the reducible fibers of π:
+Tv := Z⟨Θv,i | i ̸= 0⟩ for v ∈ R,
+T :=
+�
+v∈R
+Tv.
+By Kodaira’s classification [SS19, Thm. 5.12], each Tv with v ∈ R is represented by a Dynkin
+diagram Am, Dm or Em for some m. We also define the trivial lattice of X, namely
+Triv(X) := Z⟨O, Θv,i | i ≥ 0, v ∈ R⟩.
+Next we define the Mordell-Weil lattice and present the height formula.
+The Mordell-Weil lattice. In order to give E(K) a lattice structure, we cannot use the inter-
+section pairing directly, which only defines a lattice on NS(X) but not on E(K). This is achieved
+by defining a Q-valued pairing, called the height pairing, given by
+⟨·, ·⟩ : E(K) × E(K) → Q
+P, Q �→ −ϕ(P) · ϕ(Q),
+where ϕ : E(K) → NS(X) ⊗Z Q is defined from the orthogonal projection with respect to Triv(X)
+(for a detailed exposition, see [SS19, Section 6.5]). Moreover, dividing by torsion elements we get
+a positive-definite lattice (E(K)/E(K)tor, ⟨·, ·⟩) [SS19, Thm. 6.20], called the Mordell-Weil lattice.
+The height formula. The height pairing can be explicitly computed by the height formula [SS19,
+Thm. 6.24]. For rational elliptic surfaces, it is given by
+⟨P, Q⟩ = 1 + (P · O) + (Q · O) − (P · Q) −
+�
+v∈R
+contrv(P, Q),
+(1)
+h(P) := ⟨P, P⟩ = 2 + 2(P · O) −
+�
+v∈R
+contrv(P),
+(2)
+where contrv(P) := contrv(P, P) and contrv(P, Q) are given by Table 1 [SS19, Table 6.1] assuming
+P, Q meet π−1(v) at Θv,i, Θv,j resp. with 0 < i < j. If P or Q meets Θv,0, then contrv(P, Q) := 0.
+The minimal norm. Since E(K) is finitely generated, there is a minimal positive value for h(P)
+as P runs through E(K) with h(P) > 0. It is called the minimal norm, denoted by
+µ := min{h(P) > 0 | P ∈ E(K)}.
+The narrow Mordell-Weil lattice. An important sublattice of E(K) is the narrow Mordell-Weil
+lattice E(K)0, defined as
+E(K)0 := {P ∈ E(K) | P intersects Θv,0 for all v ∈ R}
+= {P ∈ E(K) | contrv(P) = 0 for all v ∈ R}.
+As a subgroup, E(K)0 is torsion-free; as a sublattice, it is a positive-definite even integral lattice
+with finite index in E(K) [SS19, Thm. 6.44]. The importance of the narrow lattice can be explained
+by its considerable size as a sublattice and by the easiness to compute the height pairing on it,
+since all contribution terms vanish. A complete classification of the lattices E(K) and E(K)0 on
+rational elliptic surfaces is found in [OS91, Main Thm.].
+5
+
+Tv
+A1
+E7
+A2
+E6
+An−1
+Dn+4
+Type of π−1(v)
+III
+III∗
+IV
+IV∗
+In
+I∗
+n
+contrv(P)
+1
+2
+3
+2
+2
+3
+4
+3
+i(n−i)
+n
+�
+1
+(i = 1)
+1 + n
+4
+(i > 1)
+contrv(P, Q)
+-
+-
+1
+3
+2
+3
+i(n−j)
+n
+� 1
+2
+(i = 1)
+1
+2 + n
+4
+(i > 1)
+Table 1: Local contributions from reducible fibers to the height pairing.
+2.2
+Gap numbers
+We introduce some convenient terminology to express the possibility of finding a pair of sections
+with a given intersection number.
+Definition 2.1. If there are no sections P1, P2 ∈ E(K) such that P1 · P2 = k, we say that X has
+a k-gap or that k is a gap number of X.
+Definition 2.2. We say that X is gap-free if for every k ∈ Z≥0 there are sections P1, P2 ∈ E(K)
+such that P1 · P2 = k.
+Remark 2.3. In case the Mordell-Weil rank is r = 0, we have E(K) = E(K)tor. In particular,
+any two distinct sections are disjoint [SS19, Cor. 8.30], hence every k ≥ 1 is a gap number of X.
+For positive rank, the description of gap numbers is less trivial, thus our focus on r ≥ 1.
+2.3
+Bounds cmax, cmin for the contribution term
+We define the estimates cmax, cmin for the contribution term �
+v contrv(P) and state some
+simple facts about them. We also provide an example to illustrate how they are computed.
+The need for these estimates comes from the following. Suppose we are given a section P ∈ E(K)
+whose height h(P) is known and we want to determine P · O. In case P ∈ E(K)0 we have a direct
+answer, namely P · O = h(P)/2 − 1 by the height formula (2).
+However if P /∈ E(K)0, the
+computation of P · O depends on the contribution term cP := �
+v∈R contrv(P), which by Table 1
+depends on how P intersects the reducible fibers of π. Usually we do not have this intersection
+data at hand, which is why we need estimates for cP not depending on P.
+Definition 2.4. If the set R of reducible fibers of π is not empty, we define
+cmax :=
+�
+v∈R
+max{contrv(P) | P ∈ E(K)},
+cmin := min {contrv(P) > 0 | P ∈ E(K), v ∈ R} .
+Remark 2.5. The case R = ∅ only occurs when X has Mordell-Weil rank r = 8 (No. 1 in Table 8).
+In this case E(K)0 = E(K) and �
+v∈R contrv(P) = 0 ∀P ∈ E(K), hence we adopt the convention
+cmax = cmin = 0.
+6
+
+Remark 2.6. We use cmax, cmin as bounds for cP := �
+v contrv(P). For our purposes it is not
+necessary to know whether cP actually attains one of these bounds for some P, so that cmax, cmin
+should be understood as hypothetical values.
+We state some facts about cmax, cmin.
+Lemma 2.7. Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1. If π admits a
+reducible fiber, then:
+i) cmin > 0.
+ii) cmax < 4.
+iii) cmin ≤ �
+v∈R contrv(P) ≤ cmax ∀P /∈ E(K)0. For P ∈ E(K)0, only the second inequality holds.
+iv) If �
+v∈R contrv(P) = cmin, then contrv′(P) = cmin for some v′ and contrv(P) = 0 for v ̸= v′.
+Proof. Item i) is immediate from the definition of cmin. For ii) it is enough to check the values
+of cmax directly in Table 8. For iii), the second inequality follows from the definition of cmax and
+clearly holds for any P ∈ E(K). If P /∈ E(K)0, then cP := �
+v contrv(P) > 0, so contrv0(P) > 0
+for some v0. Therefore cP ≥ contrv0(P) ≥ cmin.
+For iv), let �
+v contrv(P) = cmin. Assume by contradiction that there are distinct v1, v2 such
+that contrvi(P) > 0 for i = 1, 2. By definition of cmin we have cmin ≤ contrvi(P) for i = 1, 2 so
+cmin =
+�
+v
+contrv(P) ≥ contrv1(P) + contrv2(P) ≥ 2cmin,
+which is absurd because cmin > 0 by i). Therefore there is only one v′ with contrv′(P) > 0, while
+contrv(P) = 0 for all v ̸= v′. In particular, contrv′(P) = cmin. ■
+Explicit computation. Once we know the lattice T associated with the reducible fibers of π
+(Section 2.1), the computation of cmax, cmin is simple. For a fixed v ∈ R, the extreme values of the
+local contribution contrv(P) are given in Table 2, which is derived from Table 1. We provide an
+example to illustrate this computation.
+Tv
+max{contrv(P) | P ∈ E(K)}
+min{contrv(P) > 0 | P ∈ E(K)}
+An−1
+ℓ(n−ℓ)
+n
+, where ℓ :=
+�n
+2
+�
+n−1
+n
+Dn+4
+1 + n
+4
+1
+E6
+4
+3
+4
+3
+E7
+3
+2
+3
+2
+Table 2: Extreme values of contrv(P).
+7
+
+Example: Let π with fiber configuration (I4, IV, III, I1). The reducible fibers are I4, IV, III, so
+T = A3 ⊕ A2 ⊕ A1.
+By Table 2, the maximal contributions for A3, A2, A1 are 2·2
+4
+= 1,
+2
+3,
+1
+2
+respectively. The minimal positive contributions are 1·3
+4 = 3
+4, 2
+3, 1
+2 respectively. Then
+cmax = 1 + 2
+3 + 1
+2 = 13
+6 ,
+cmin = min
+�3
+4, 2
+3, 1
+2
+�
+= 1
+2.
+2.4
+The difference ∆ = cmax − cmin
+In this section we explain why the value of ∆ := cmax − cmin is relevant to our discussion,
+specially in Subsection 4.2. We also verify that ∆ < 2 in most cases and identify the exceptional
+ones in Table 3 and Table 4.
+As noted in Subsection 2.3, in case P /∈ E(K)0 and h(P) is known, the difficulty of determining
+P ·O lies in the contribution term cP := �
+v∈R contrv(P). In particular, the range of possible values
+for cP determines the possibilities for P · O. This range is measured by the difference
+∆ := cmax − cmin.
+Hence a smaller ∆ means a better control over the intersection number P · O, which is why ∆
+plays an important role in determining possible intersection numbers. In Subsection 4.3 we assume
+∆ ≤ 2 and state necessary and sufficient conditions for having a pair P1, P2 such that P1 · P2 = k
+for a given k ≥ 0. If however ∆ > 2, the existence of such a pair is not guaranteed a priori, so a
+case-by-case treatment is needed. Fortunately by Lemma 2.8 the case ∆ > 2 is rare.
+Lemma 2.8. Let X be a rational elliptic surface with Mordell-Weil rank r ≥ 1. The only cases
+with ∆ = 2 and ∆ > 2 are in Table 3 and 4 respectively. In particular we have ∆ < 2 whenever
+E(K) is torsion-free.
+No.
+T
+E(K)
+cmax
+cmin
+24
+A⊕5
+1
+A∗
+1
+⊕3 ⊕ Z/2Z
+5
+2
+1
+2
+38
+A3 ⊕ A⊕3
+1
+A∗
+1 ⊕ ⟨1/4⟩ ⊕ Z/2Z
+5
+2
+1
+2
+53
+A5 ⊕ A⊕2
+1
+⟨1/6⟩ ⊕ Z/2Z
+5
+2
+1
+2
+57
+D4 ⊕ A⊕3
+1
+A∗
+1 ⊕ (Z/2Z)⊕2
+5
+2
+1
+2
+58
+A⊕2
+3
+⊕ A1
+A∗
+1 ⊕ Z/4Z
+5
+2
+1
+2
+61
+A⊕3
+2
+⊕ A1
+⟨1/6⟩ ⊕ Z/3Z
+5
+2
+1
+2
+Table 3: Cases with ∆ = 2
+8
+
+No.
+T
+E(K)
+cmax
+cmin
+∆
+41
+A2 ⊕ A⊕4
+1
+1
+6
+�
+2
+1
+1
+2
+�
+⊕ Z/2Z
+8
+3
+1
+2
+13
+6
+42
+A⊕6
+1
+A∗
+1
+⊕2 ⊕ (Z/2Z)⊕2
+3
+1
+2
+5
+2
+59
+A3 ⊕ A2 ⊕ A⊕2
+1
+⟨1/12⟩ ⊕ Z/2Z
+8
+3
+1
+2
+13
+6
+60
+A3 ⊕ A⊕4
+1
+⟨1/4⟩ ⊕ (Z/2Z)⊕2
+3
+1
+2
+5
+2
+Table 4: Cases with ∆ > 2
+Proof. By searching Table 8 for all cases with ∆ = 2 and ∆ > 2, we obtain Table 3 and Table 4
+respectively. Notice in particular that in both tables the torsion part of E(K) is always nontrivial.
+Consequently, if E(K) is torsion-free, then ∆ < 2. ■
+2.5
+The quadratic form QX
+We define the positive-definite quadratic form with integer coefficients QX derived from the
+height pairing. The relevance of QX is due to the fact that some conditions for having P1 · P2 = k
+for some P1, P2 ∈ E(K) can be stated in terms of what integers can be represented by QX (see
+Corollary 4.2 and Proposition 4.12).
+The definition of QX consists in clearing denominators of the rational quadratic form induced
+by the height pairing; the only question is how to find a scale factor that works in every case. More
+precisely, if E(K) has rank r ≥ 1 and P1, ..., Pr are generators of its free part, then q(x1, ..., xr) :=
+h(x1P1 + ... + xrPr) is a quadratic form with coefficients in Q; we define QX by multiplying q by
+some integer d > 0 so as to produce coefficients in Z. We show that d may always be chosen as the
+determinant of the narrow lattice E(K)0.
+Definition 2.9. Let X with r ≥ 1. Let P1, ..., Pr be generators of the free part of E(K). Define
+QX(x1, ..., xr) := (det E(K)0) · h(x1P1 + ... + xrPr).
+We check that the matrix representing QX has entries in Z, therefore QX has coefficients in Z.
+Lemma 2.10. Let A be the matrix representing the quadratic form QX, i.e. Q(x1, ..., xr) = xtAx,
+where x := (x1, ..., xr)t. Then A has integer entries. In particular, QX has integer coefficients.
+Proof. Let P1, ..., Pr be generators of the free part of E(K) and let L := E(K)0. The free part of
+E(K) is isomorphic to the dual lattice L∗ [OS91, Main Thm.], so we may find generators P 0
+1 , ..., P 0
+r
+of L such that the Gram matrix B0 := (⟨P 0
+i , P 0
+j ⟩)i,j of L is the inverse of the Gram matrix
+B := (⟨Pi, Pj⟩)i,j of L∗.
+9
+
+We claim that QX is represented by the adjugate matrix of B0, i.e. the matrix adj(B0) such
+that B0 · adj(B0) = (det B0) · Ir, where Ir is the r × r identity matrix. Indeed, by construction B
+represents the quadratic form h(x1P1 + ... + xrPr), therefore
+QX(x1, ..., xr) = (det E(K)0) · h(x1P1 + ... + xrPr)
+= (det B0) · xtBx
+= (det B0) · xt(B0)−1x
+= xtadj(B0)x,
+as claimed. To prove that A := adj(B0) has integer coefficients, notice that the Gram matrix
+B0 of L = E(K)0 has integer coefficients (as E(K)0 is an even lattice), then so does A. ■
+We close this subsection with a simple consequence of the definition of QX.
+Lemma 2.11. If h(P) = m for some P ∈ E(K), then QX represents d · m, where d := det E(K)0.
+Proof. Let P1, ..., Pr be generators for the free part of E(K). Let P = a1P1 + ... + arPr + Q, where
+ai ∈ Z and Q is a torsion element (possibly zero). Since torsion sections do not contribute to the
+height pairing, then h(P − Q) = h(P) = m. Hence
+QX(a1, ..., ar) = d · h(a1P1 + ... + arPr)
+= d · h(P − Q)
+= d · m. ■
+3
+Intersection with a torsion section
+Before dealing with more technical details in Section 4, we explain how torsion sections can be
+of help in our investigation, specially in Subsection 4.2.
+We first note some general properties of torsion sections. As the height pairing is positive-
+definite on E(K)/E(K)tor, torsion sections are inert in the sense that for each Q ∈ E(K)tor we
+have ⟨Q, P⟩ = 0 for all P ∈ E(K).
+Moreover, in the case of rational elliptic surfaces, torsion
+sections also happen to be mutually disjoint:
+Theorem 3.1. [MP89, Lemma 1.1] On a rational elliptic surface, Q1 · Q2 = 0 for any distinct
+Q1, Q2 ∈ E(K)tor. In particular, if O is the neutral section, then Q·O = 0 for all Q ∈ E(K)tor\{O}.
+Remark 3.2. As stated in [MP89, Lemma 1.1], Theorem 3.1 holds for elliptic surfaces over C even
+without assuming X is rational. However, for an arbitrary algebraically closed field the rationality
+hypothesis is needed, and a proof can be found in [SS19, Cor. 8.30].
+By taking advantage of the properties above, we use torsion sections to help us find P1, P2 ∈
+E(K) such that P1 · P2 = k for a given k ∈ Z≥0. This is particularly useful when ∆ ≥ 2, in which
+case E(K)tor is not trivial by Lemma 2.8.
+The idea is as follows. Given k ∈ Z≥0, suppose we can find P ∈ E(K)0 with height h(P) = 2k.
+By the height formula (2), P · O = k − 1 < k, which is not yet what we need. In the next lemma
+we show that replacing O with a torsion section Q ̸= O gives P · Q = k, as desired.
+10
+
+Lemma 3.3. Let P ∈ E(K)0 such that h(P) = 2k. Then P · Q = k for all Q ∈ E(K)tor \ {O}.
+Proof. Assume there is some Q ∈ E(K)tor \ {O}. By Theorem 3.1, Q · O = 0 and by the height
+formula (2), 2k = 2 + 2(P · O) − 0, hence P · O = k − 1. We use the height formula (1) for ⟨P, Q⟩
+in order to conclude that P · Q = k. Since P ∈ E(K)0, it intersects the neutral component Θv,0 of
+every reducible fiber π−1(v), so contrv(P, Q) = 0 for all v ∈ R. Hence
+0 = ⟨P, Q⟩
+= 1 + P · O + Q · O − P · Q −
+�
+v∈R
+contrv(P, Q)
+= 1 + (k − 1) + 0 − P · Q − 0
+= k − P · Q. ■
+4
+Existence of a pair of sections with a given intersection number
+Given k ∈ Z≥0, we state necessary and (in most cases) sufficient conditions for having
+P1 ·P2 = k for some P1, P2 ∈ E(K). Necessary conditions are stated in generality in Subsection 4.1,
+while sufficient ones depend on the value of ∆ and are treated separately in Subsection 4.2. In
+Subsection 4.4, we collect all sufficient conditions proven in this section.
+4.1
+Necessary Conditions
+If k ∈ Z≥0, we state necessary conditions for having P1·P2 = k for some sections P1, P2 ∈ E(K).
+We note that the value of ∆ is not relevant in this subsection, although it plays a decisive role for
+sufficient conditions in Subsection 4.2.
+Lemma 4.1. Let k ∈ Z≥0. If P1 · P2 = k for some P1, P2 ∈ E(K), then one of the following holds:
+i) h(P) = 2 + 2k for some P ∈ E(K)0.
+ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some P /∈ E(K)0.
+Proof. Without loss of generality we may assume P2 is the neutral section, so that P1 · O = k. By
+the height formula (2), h(P1) = 2 + 2k − c, where c := �
+v contrv(P1). If P1 ∈ E(K)0, then c = 0
+and h(P1) = 2 + 2k, hence i) holds. If P1 /∈ E(K)0, then cmin ≤ c ≤ cmax by Lemma 2.7. But
+h(P1) = 2 + 2k − c, therefore 2 + 2k − cmax ≤ h(P1) ≤ 2 + 2k − cmin, i.e. ii) holds. ■
+Corollary 4.2. Let k ∈ Z≥0. If P1 · P2 = k for some P1, P2 ∈ E(K), then QX represents some
+integer in [d · (2 + 2k − cmax), d · (2 + 2k)], where d := det E(K)0.
+Proof.
+We apply Lemma 4.1 and rephrase it in terms of QX. If i) holds, then QX represents
+d · (2 + 2k) by Lemma 2.11. But if ii) holds, then h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] and by
+Lemma 2.11, QX represents d · h(P) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)]. In both i) and ii),
+QX represents some integer in [d · (2 + 2k − cmax), d · (2 + 2k)]. ■
+11
+
+4.2
+Sufficient conditions when ∆ ≤ 2
+In this subsection we state sufficient conditions for having P1 · P2 = k for some P1, P2 ∈ E(K)
+under the assumption that ∆ ≤ 2. By Lemma 2.8, this covers almost all cases (more precisely, all
+but No. 41, 42, 59, 60 in Table 8). We treat ∆ < 2 and ∆ = 2 separately, as the latter needs more
+attention.
+4.2.1
+The case ∆ < 2
+We first prove Lemma 4.3, which gives sufficient conditions assuming ∆ < 2, then Corollary 4.5,
+which states sufficient conditions in terms of integers represented by QX.
+This is followed by
+Corollary 4.6, which is a simplified version of Corollary 4.5.
+Lemma 4.3. Assume ∆ < 2 and let k ∈ Z≥0. If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin] for some
+P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).
+Proof. Let O ∈ E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P · O) − c,
+where c := �
+v contrv(P). Since h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin], then
+2 + 2k − cmax ≤ 2 + 2(P · O) − c ≤ 2 + 2k − cmin
+⇒ c − cmax
+2
+≤ P · O − k ≤ c − cmin
+2
+.
+Therefore P · O − k is an integer in I :=
+� c−cmax
+2
+, c−cmin
+2
+�. We prove that 0 is the only integer in
+I, so that P · O − k = 0, i.e. P · O = k. First notice that c ̸= 0, as P /∈ E(K)0. By Lemma 2.7 iii),
+cmin ≤ c ≤ cmax, consequently c−cmax
+2
+≤ 0 ≤ c−cmin
+2
+, i.e. 0 ∈ I. Moreover ∆ < 2 implies that I has
+length cmax−cmin
+2
+= ∆
+2 < 1, so I contains no integer except 0 as desired. ■
+Remark 4.4. Lemma 4.3 also applies when cmax = cmin, in which case the closed interval degen-
+erates into a point.
+The following corollary of Lemma 4.3 states a sufficient condition in terms of integers represented
+by the quadratic form QX (Section 2.5).
+Corollary 4.5. Assume ∆ < 2 and let d := det E(K)0. If QX represents an integer not divisible
+by d in the interval [d · (2+ 2k − cmax), d · (2+ 2k − cmin)], then P1 · P2 = k for some P1, P2 ∈ E(K).
+Proof. Let a1, ..., ar ∈ Z such that QX(a1, ..., ar) ∈ [d · (2 + 2k − cmax), d · (2 + 2k − cmin)] with
+d ∤ QX(a1, ..., ar). Let P := a1P1 + ... + arPr, where P1, ..., Pr are generators of the free part of
+E(K). Then d ∤ QX(a1, ..., ar) = d · h(P), which implies that h(P) /∈ Z. In particular P /∈ E(K)0
+since E(K)0 is an integer lattice. Moreover h(P) = 1
+dQX(a1, ..., ar) ∈ [2 + 2k − cmax, 2 + 2k − cmin]
+and we are done by Lemma 4.3. ■
+12
+
+The next corollary, although weaker than Corollary 4.5, is more practical for concrete examples
+and is frequently used in Subsection 5.4. It does not involve finding integers represented by QX,
+but only finding perfect squares in an interval depending on the minimal norm µ (Subsection 2.1).
+Corollary 4.6. Assume ∆ < 2. If there is a perfect square n2 ∈
+�
+2+2k−cmax
+µ
+, 2+2k−cmin
+µ
+�
+such that
+n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K).
+Proof. Take P ∈ E(K) such that h(P) = µ. Since h(nP) = n2µ /∈ Z, we must have nP /∈ E(K)0
+as E(K)0 is an integer lattice. Moreover h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin] and we are
+done by Lemma 4.3. ■
+4.2.2
+The case ∆ = 2
+The statement of sufficient conditions for ∆ = 2 is almost identical to the one for ∆ < 2: the
+only difference is that the closed interval Lemma 4.3 is substituted by a right half-open interval
+in Lemma 4.8. This change, however, is associated with a technical difficulty in the case when a
+section has minimal contribution term, thus the separate treatment for ∆ = 2.
+The results are presented in the following order. First we prove Lemma 4.7, which is a statement
+about sections whose contribution term is minimal.
+Next we prove Lemma 4.8, which states
+sufficient conditions for ∆ = 2, then Corollaries 4.9 and 4.10.
+Lemma 4.7. Assume ∆ = 2.
+If there is P ∈ E(K) such that �
+v∈R contrv(P) = cmin, then
+P · Q = P · O + 1 for every Q ∈ E(K)tor \ {O}.
+Proof. If Q ∈ E(K)tor \ {O}, then Q · O = 0 by Theorem 3.1. Moreover, by the height formula (1),
+0 = ⟨P, Q⟩ = 1 + P · O + 0 − P · Q −
+�
+v∈R
+contrv(P, Q). (∗)
+Hence it suffices to show that contrv(P, Q) = 0 ∀v ∈ R. By Lemma 2.7 iv), contrv′(P) = cmin
+for some v′ and contrv(P) = 0 for all v ̸= v′. In particular P meets Θv,0, hence contrv(P, Q) = 0
+for all v ̸= v′. Thus from (∗) we see that contrv′(P, Q) is an integer, which we prove is 0.
+We claim that Tv′ = A1, so that contrv′(P, Q) = 0 or 1
+2 by Table 1. In this case, as contrv′(P, Q)
+is an integer, it must be 0, and we are done. To see that Tv′ = A1 we analyse contrv′(P). Since
+∆ = 2, then cmin = 1
+2 by Table 3 and contrv′(P) = cmin = 1
+2. By Table 1, this only happens if
+Tv′ = An−1 and 1
+2 = i(n−i)
+n
+for some 0 ≤ i < n. The only possibility is i = 1, n = 2 and Tv′ = A1. ■
+With the aid of Lemma 4.7 we are able to state sufficient conditions for ∆ = 2.
+13
+
+Lemma 4.8. Assume ∆ = 2 and let k ∈ Z≥0. If h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some
+P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).
+Proof. Let O ∈ E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P · O) − c,
+where c := �
+v contrv(P). We repeat the arguments from Lemma 4.3, in this case with the right
+half-open interval, so that the hypothesis that h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin), implies that
+P · O − k is an integer in I′ :=
+� c−cmax
+2
+, c−cmin
+2
+�.
+Since I′ is half-open with length cmax−cmin
+2
+= ∆
+2 = 1, then I′ contains exactly one integer. If
+0 ∈ I′, then P · O − k = 0, i.e. P · O = k and we are done. Hence we assume 0 /∈ I′.
+We claim that P ·O = k −1. First, notice that if c > cmin, then the inequalities cmin < c ≤ cmax
+give c−cmax
+2
+≤ 0 < c−cmin
+2
+, i.e. 0 ∈ I′, which is a contradiction. Hence c = cmin. Since ∆ = 2, then
+I′ = [−1, 0), whose only integer is −1. Thus P · O − k = −1, i.e. P · O = k − 1, as claimed.
+Finally, let Q ∈ E(K)tor \ {O}, so that P · Q = P · O + 1 = k by Lemma 4.7 and we are done.
+We remark that E(K)tor is not trivial by Table 3, therefore such Q exists. ■
+The following corollaries are analogues to Corollary 4.5 and Corollary 4.6 adapted to ∆ = 2.
+Similarly to the case ∆ < 2, Corollary 4.9 is stronger than Corollary 4.10, although the latter is
+more practical for concrete examples. We remind the reader that µ denotes the minimal norm
+(Subsection 2.1).
+Corollary 4.9. Assume ∆ = 2 and let d := det E(K)0. If QX represents an integer not divisible
+by d in the interval [d·(2+2k −cmax), d·(2+2k −cmin)), then P1 ·P2 = k for some P1, P2 ∈ E(K).
+Proof. We repeat the arguments in Corollary 4.5, in this case with the half-open interval. ■
+Corollary 4.10. Assume ∆ = 2. If there is a perfect square n2 ∈
+�
+2+2k−cmax
+µ
+, 2+2k−cmin
+µ
+�
+such that
+n2µ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K).
+Proof. We repeat the arguments in Corollary 4.6, in this case with the half-open interval. ■
+4.3
+Necessary and sufficient conditions for ∆ ≤ 2
+For completeness, we present a unified statement of necessary and sufficient conditions assuming
+∆ ≤ 2, which follows naturally from results in Subsections 4.1 and 4.2.
+Lemma 4.11. Assume ∆ ≤ 2 and let k ∈ Z≥0. Then P1 · P2 = k for some P1, P2 ∈ E(K) if and
+only if one of the following holds:
+i) h(P) = 2 + 2k for some P ∈ E(K)0.
+ii) h(P) ∈ [2 + 2k − cmax, 2 + 2k − cmin) for some P /∈ E(K)0.
+iii) h(P) = 2 + 2k − cmin and �
+v∈R contrv(P) = cmin for some P ∈ E(K).
+Proof. If i) or iii) holds, then P · O = k directly by the height formula (2). But if ii) holds, it
+suffices to to apply Lemma 4.3 when ∆ < 2 and by Lemma 4.8 when ∆ = 2.
+Conversely, let P1·P2 = k. Without loss of generality, we may assume P2 = O, so that P1·O = k.
+By the height formula (2), h(P1) = 2 + 2k − c, where c := �
+v contrv(P1).
+If c = 0, then P1 ∈ E(K)0 and h(P1) = 2+2k, so i) holds. Hence we let c ̸= 0, i.e. P1 /∈ E(K)0,
+so that cmin ≤ c ≤ cmax by Lemma 2.7. In case c = cmin, then h(P1) = 2 + 2k − cmin and iii) holds.
+Otherwise cmin < c ≤ cmax, which implies 2 + 2k − cmax ≤ h(P1) < 2 + 2k − cmin, so ii) holds. ■
+14
+
+4.4
+Summary of sufficient conditions
+For the sake of clarity, we summarize in a single proposition all sufficient conditions for having
+P1 · P2 = k for some P1, P2 ∈ E(K) proven in this section.
+Proposition 4.12. Let k ∈ Z≥0. If one of the following holds, then P1 · P2 = k for some P1, P2 ∈
+E(K).
+1) h(P) = 2 + 2k for some P ∈ E(K)0.
+2) h(P) = 2k for some P ∈ E(K)0 and E(K)tor is not trivial.
+3) ∆ < 2 and there is a perfect square n2 ∈
+�
+2+2k−cmax
+µ
+, 2+2k−cmin
+µ
+�
+with n2µ /∈ Z, where µ is the
+minimal norm (Subsection 2.1). In case ∆ = 2, consider the right half-open interval.
+4) ∆ < 2 and the quadratic form QX represents an integer not divisible by d := det E(K)0 in the
+interval [d · (2 + 2k − cmax), d · (2 + 2k − cmin)]. In case ∆ = 2, consider the right half-open
+interval.
+Proof. In 1) a height calculation gives 2 + 2k = h(P) = 2 + 2(P · O) − 0, so P · O = k. For
+2), we apply Lemma 3.3 to conclude that P · Q = k for any Q ∈ E(K)tor \ {O}. In 3) we use
+Corollary 4.6 when ∆ < 2 and Corollary 4.10 when ∆ = 2. In 4), we apply Corollary 4.5 if ∆ < 2
+and Corollary 4.9 if ∆ = 2. ■
+5
+Main Results
+We prove the four main theorems of this paper, which are independent applications of the results
+from Section 4. The first two are general attempts to describe when and how gap numbers occur:
+Theorem 5.2 tells us that large Mordell-Weil groups prevent the existence of gaps numbers, more
+precisely for Mordell-Weil rank r ≥ 5; in Theorem 5.4 we show that for small Mordell-Weil rank,
+more precisely when r ≤ 2, then gap numbers occur with probability 1. The last two theorems,
+on the other hand, deal with explicit values of gap numbers: Theorem 5.7 provides a complete
+description of gap numbers in certain cases, while Theorem 5.8 is a classification of cases with a
+1-gap.
+5.1
+No gap numbers in rank r ≥ 5
+We show that if E(K) has rank r ≥ 5, then X is gap-free. Our strategy is to prove that for
+every k ∈ Z≥0 there is some P ∈ E(K)0 such that h(P) = 2+2k, and by Proposition 4.12 1) we are
+done. We accomplish this in two steps. First we show that this holds when there is an embedding
+of A⊕
+1 or of A4 in E(K)0 (Lemma 5.1). Second, we show that if r ≥ 5, then such embedding exists,
+hence X is gap-free (Theorem 5.2).
+15
+
+Lemma 5.1. Assume E(K)0 has a sublattice isomorphic to A⊕4
+1
+or A4. Then for every ℓ ∈ Z≥0
+there is P ∈ E(K)0 such that h(P) = 2ℓ.
+Proof.
+First assume A⊕4
+1
+⊂ E(K)0 and let P1, P2, P3, P4 be generators for each factor A1 in
+A⊕4
+1 . Then h(Pi) = 2 and ⟨Pi, Pj⟩ = 0 for distinct i, j = 1, 2, 3, 4.
+By Lagrange’s four-square
+theorem [HW79, §20.5] there are integers a1, a2, a3, a4 such that a2
+1 + a2
+2 + a2
+3 + a2
+4 = ℓ. Defining
+P := a1P1 + a2P2 + a3P3 + a4P4 ∈ A⊕4
+1
+⊂ E(K)0, we have
+h(P) = 2a2
+1 + 2a2
+2 + 2a2
+3 + 2a2
+4 = 2ℓ.
+Now let A4 ⊂ E(K)0 with generators P1, P2, P3, P4.
+Then h(Pi) = 2 for i = 1, 2, 3, 4 and
+⟨Pi, Pi+1⟩ = −1 for i = 1, 2, 3. We need to find integers x1, ..., x4 such that h(P) = 2ℓ, where
+P := x1P1 + ... + x4P4 ∈ A4 ⊂ E(K)0. Equivalently, we need that
+ℓ = 1
+2⟨P, P⟩ = x2
+1 + x2
+2 + x2
+3 + x2
+4 − x1x2 − x2x3 − x3x4.
+Therefore ℓ must be represented by q(x1, ..., x4) := x2
+1 + x2
+2 + x2
+3 + x2
+4 − x1x2 − x2x3 − x3x4. We
+prove that q represents all positive integers. Notice that q is positive-definite, since it is induced
+by ⟨·, ·⟩. By Bhargava-Hanke’s 290-theorem [BH][Thm. 1], q represents all positive integers if and
+only if it represents the following integers:
+2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26,
+29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290.
+The representation for each of the above is found in Table 5. ■
+We now prove the main theorem of this section.
+Theorem 5.2. If r ≥ 5, then X is gap-free.
+Proof. We show that for every k ≥ 0 there is P ∈ E(K)0 such that h(P) = 2 + 2k, so that by
+Proposition 4.12 1) we are done. Using Lemma 5.1 it suffices to prove that E(K)0 has a sublattice
+isomorphic to A⊕4
+1
+or A4.
+The cases with r ≥ 5 are No.
+1-7 (Table 8).
+In No.
+1-6, E(K)0 = E8, E7, E6, D6, D5, A5
+respectively. Each of these admit an A4 sublattice [Nis96, Lemmas 4.2,4.3]. In No. 7 we claim that
+E(K)0 = D4 ⊕ A1 has an A⊕4
+1
+sublattice. This is the case because D4 admits an A⊕4
+1
+sublattice
+[Nis96, Lemma 4.5 (iii)]. ■
+16
+
+n
+x1, x2, x3, x4 with x2
+1 + x2
+2 + x2
+3 + x2
+4 − x1x2 − x2x3 − x3x4 = n
+1
+1, 0, 0, 0
+2
+1, 0, 1, 0
+3
+1, 1, 2, 0
+5
+1, 0, 2, 0
+6
+1, 1, −2, −1
+7
+1, 1, −2, 0
+10
+1, 0, 3, 0
+13
+2, 0, 3, 0
+14
+1, 2, 5, 1
+15
+1, 5, 5, 2
+17
+1, 0, 4, 0
+19
+1, 5, 3, −1
+21
+1, 5, 0, 0
+22
+1, 5, 0, −1
+23
+1, 6, 6, 2
+26
+1, 0, 5, 0
+29
+2, 0, 5, 0
+30
+1, 5, 0, −3
+31
+1, 3, −4, −2
+34
+3, 0, 5, 0
+35
+1, 2, −2, 4
+37
+1, 0, 6, 0
+42
+1, 1, −4, 3
+58
+3, 0, 7, 0
+93
+1, 1, −10, 0
+110
+1, −2, 3, −8
+145
+1, 0, 12, 0
+203
+1, −5, −9, 8
+290
+1, 0, 17, 0
+Table 5: Representation of the critical integers in Bhargava-Hanke’s 290-theorem.
+5.2
+Gaps with probability 1 in rank r ≤ 2
+Fix a rational elliptic surface π : X → P1 with Mordell-Weil rank r ≤ 2. We prove that if k is
+a uniformly random natural number, then k is a gap number with probability 1. More precisely, if
+G := {k ∈ N | k is a gap number of X} is the set of gap numbers, then G ⊂ N has density 1, i.e.
+d(G) := lim
+n→∞
+#G ∩ {1, ..., n}
+n
+= 1.
+17
+
+We adopt the following strategy. If k ∈ N \ G, then P1 · P2 = k for some P1, P2 ∈ E(K) and
+by Corollary 4.2 the quadratic form QX represents some integer t depending on k. This defines a
+function N\G → T, where T is the set of integers represented by QX. Since QX is a quadratic form
+on r ≤ 2 variables, T has density 0 in N by Lemma 5.3. By analyzing the pre-images of N\G → T,
+in Theorem 5.4 we conclude that d(N \ G) = d(T) = 0, hence d(G) = 1 as desired.
+Lemma 5.3. Let Q be a positive-definite quadratic form on r = 1, 2 variables with integer coeffi-
+cients. Then the set of integers represented by Q has density 0 in N.
+Proof. Let S be the set of integers represented by Q. If d is the greatest common divisor of the
+coefficients of Q, let S′ be the set of integers representable by the primitive form Q′ := 1
+d · Q. By
+construction S′ is a rescaling of S, so d(S) = 0 if and only if d(S′) = 0.
+If r = 1, then Q′(x1) = x2
+1 and S′ is the set of perfect squares, so clearly d(S′) = 0. If r = 2,
+then Q′ is a binary quadratic form and the number of elements in S′ bounded from above by x > 0
+is given by C ·
+x
+√log x + o(x) with C > 0 a constant and limx→∞
+o(x)
+x
+= 0 [Ber12, p. 91]. Thus
+d(S′) = lim
+x→∞
+C
+√log x + o(x)
+x
+= 0. ■
+We now prove the main result of this section.
+Theorem 5.4. Let π : X → P1 be a rational elliptic surface with Mordell-Weil rank r ≤ 2. Then
+the set G := {k ∈ N | k is a gap number of X} of gap numbers of X has density 1 in N.
+Proof. If r = 0, then the claim is trivial by Remark 2.3, hence we may assume r = 1, 2. We
+prove that S := N \ G has density 0.
+If S is finite, there is nothing to prove.
+Otherwise, let
+k1 < k2 < ... be the increasing sequence of all elements of S. By Corollary 4.2, for each n there is
+some tn ∈ Jkn := [d · (2 + 2kn − cmax), d · (2 + 2kn)] represented by the quadratic form QX. Let T
+be the set of integers represented by QX and define the function f : N \ G → T by kn �→ tn. Since
+QX has r = 1, 2 variables, T has density 0 by Lemma 5.3.
+For N > 0, let SN := S ∩ {1, ..., N} and TN := T ∩ {1, ..., N}.
+Since T has density zero,
+#TN = o(N), i.e.
+#TN
+N
+→ 0 when N → ∞ and we need to prove that #SN = o(N). We analyze
+the function f restricted to SN. Notice that as tn ∈ Jkn, then kn ≤ N implies tn ≤ d · (2 + 2kn) ≤
+d · (2 + 2N). Hence the restriction g := f|SN can be regarded as a function g : SN → Td·(2+2k).
+We claim that #g−1(t) ≤ 2 for all t ∈ Td·(2+2N), in which case #SN ≤ 2 · #Td·(2+2N) = o(N)
+and we are done. Assume by contradiction that g−1(t) contains three distinct elements, say kℓ1 <
+kℓ2 < kℓ3 with t = tℓ1 = tℓ2 = tℓ3. Since tℓi ∈ Jkℓi for each i = 1, 2, 3, then t ∈ Jkℓ1 ∩ Jkℓ2 ∩ Jkℓ3. We
+prove that Jkℓ1 and Jkℓ3 are disjoint, which yields a contradiction. Indeed, since kℓ1 < kℓ2 < kℓ3,
+in particular kℓ3 − kℓ1 ≥ 2, therefore d · (2 + 2kℓ1) ≤ d · (2 + 2kℓ3 − 4). But cmax < 4 by Lemma 2.7,
+so d · (2 + 2kℓ1) < d · (2 + 2kℓ3 − cmax), i.e. max Jkℓ1 < min Jkℓ3. Thus Jkℓ1 ∩ Jkℓ3 = ∅, as desired. ■
+5.3
+Identification of gaps when E(K) is torsion-free with rank r = 1
+The results in Subsections 5.1 and 5.2 concern the existence and the distribution of gap num-
+bers. In the following subsections we turn our attention to finding gap numbers explicitly. In this
+subsection we give a complete description of gap numbers assuming E(K) is torsion-free with rank
+r = 1. Such descriptions are difficult in the general case, but our assumption guarantees that each
+18
+
+E(K), E(K)0 is generated by a single element and that ∆ < 2 by Lemma 2.8, which makes the
+problem more accessible.
+We organize this subsection as follows. First we point out some trivial facts about generators
+of E(K), E(K)0 when r = 1 in Lemma 5.5. Next we state necessary and sufficient conditions for
+having P1 · P2 = k when E(K) is torsion-free with r = 1 in Lemma 5.6. As an application of the
+latter, we prove Theorem 5.7, which is the main result of the subsection.
+Lemma 5.5. Let X be a rational elliptic surface with Mordell-Weil rank r = 1. If P generates the
+free part of E(K), then
+a) h(P) = µ.
+b) 1/µ is an even integer.
+c) E(K)0 is generated by P0 := (1/µ)P and h(P0) = 1/µ.
+Proof. Item a) is clear. Items b), c) follow from the fact that E(K)0 is an even lattice and that
+E(K) ≃ L∗ ⊕ E(K)tor, where L := E(K)0 [OS91, Main Thm.]. ■
+In what follows we use Lemma 5.5 and results from Section 4 to state necessary and sufficient
+conditions for having P1 · P2 = k for some P1, P2 ∈ E(K) in case E(K) is torsion-free with r = 1.
+Lemma 5.6. Assume E(K) is torsion-free with rank r = 1. Then P1 · P2 = k for some P1, P2 ∈
+E(K) if and only if one of the following holds:
+i) µ · (2 + 2k) is a perfect square.
+ii) There is a perfect square n2 ∈
+�
+2+2k−cmax
+µ
+, 2+2k−cmin
+µ
+�
+such that µ · n /∈ Z.
+Proof. By Lemma 5.5, E(K) is generated by some P with h(P) = µ and E(K)0 is generated by
+P0 := n0P, where n0 := 1
+µ ∈ 2Z.
+First assume that P1·P2 = k for some P1, P2. Without loss of generality we may assume P2 = O.
+Let P1 = nP for some n ∈ Z. We show that P1 ∈ E(K)0 implies i) while P1 /∈ E(K)0 implies ii).
+If P1 ∈ E(K)0, then n0 | n, hence P1 = nP = mP0, where m :=
+n
+n0. By the height formula (2),
+2 + 2k = h(P1) = h(mP0) = m2 · 1
+µ. Hence µ · (2 + 2k) = m2, i.e. i) holds.
+If P1 /∈ E(K)0, then n0 ∤ n, hence µ · n =
+n
+n0 /∈ Z. Moreover, h(P1) = n2h(P) = n2µ and by
+the height formula (2), n2µ = h(P) = 2 + 2k − c, where c := �
+v contrv(P1) ̸= 0. The inequalities
+cmin ≤ c ≤ cmax then give 2+2k−cmax
+µ
+≤ n2 ≤ 2+2k−cmin
+µ
+. Hence ii) holds.
+Conversely, assume i) or ii) holds. Since E(K) is torsion-free, ∆ < 2 by Lemma 2.8, so we may
+apply Lemma 4.3. If i) holds, then µ · (2 + 2k) = m2 for some m ∈ Z. Since mP0 ∈ E(K)0 and
+h(mP0) =
+m2
+µ
+= 2 + 2k, we are done by Lemma 4.3 i).
+If ii) holds, the condition µ · n /∈ Z
+is equivalent to n0 ∤ n, hence nP
+/∈ E(K)0.
+Moreover n2 ∈
+�
+2+2k−cmax
+µ
+, 2+2k−cmin
+µ
+�
+, implies
+h(nP) = n2µ ∈ [2 + 2k − cmax, 2 + 2k − cmin]. By Lemma 4.3 ii), we are done. ■
+By applying Lemma 5.6 to all possible cases where E(K) is torsion-free with rank r = 1,
+we obtain the main result of this subsection.
+19
+
+Theorem 5.7. If E(K) is torsion-free with rank r = 1, then all the gap numbers of X are described
+in Table 6.
+No.
+T
+k is a gap number ⇔ none of
+the following are perfect squares
+first gap numbers
+43
+E7
+k + 1, 4k + 1
+1, 4
+45
+A7
+k+1
+4 , 16k, ..., 16k + 9
+8, 11
+46
+D7
+k+1
+2 , 8k + 1, ..., 8k + 4
+2, 5
+47
+A6 ⊕ A1
+k+1
+7 , 28k − 3, ..., 28k + 21
+12, 16
+49
+E6 ⊕ A1
+k+1
+3 , 12k + 1, ..., 12k + 9
+3, 7
+50
+D5 ⊕ A2
+k+1
+6 , 24k + 1, ..., 24k + 16
+6, 11
+55
+A4 ⊕ A3
+k+1
+10 , 40k − 4, ..., 40k + 25
+16, 20
+56
+A4 ⊕ A2 ⊕ A1
+k+1
+15 , 60k − 11, ..., 60k + 45
+22, 27
+Table 6: Description of gap numbers when E(K) is torsion-free with r = 1.
+Proof. For the sake of brevity we restrict ourselves to No. 55. The other cases are treated similarly.
+Here cmax = 2·3
+5 + 2·2
+4 = 11
+5 , cmin = min
+�
+4
+5, 3
+4
+�
+= 3
+4 and µ = 1/20.
+By Lemma 5.6, k is a gap number if and only if neither i) nor ii) occurs. Condition i) is that
+2+2k
+20
+= k+1
+10
+is a perfect square. Condition ii) is that
+�
+2+2k−cmax
+µ
+, 2+2k−cmin
+µ
+�
+= [40k − 4, 40k + 25]
+contains some n2 with 20 ∤ n. We check that 20 ∤ n for every n such that n2 = 40k + ℓ, with
+ℓ = −4, ..., 25. Indeed, if 20 | n, then 400 | n2 and in particular 40 | n2. Then 40 | (n2 − 40k) = ℓ,
+which is absurd. ■
+5.4
+Surfaces with a 1-gap
+In Subsection 5.3 we take each case in Table 6 and describe all its gap numbers.
+In this
+subsection we do the opposite, which is to fix a number and describe all cases having it as a gap
+number. We remind the reader that our motivating problem (Section 1) was to determine when
+there are sections P1, P2 such that P1 · P2 = 1, which induce a conic bundle having P1 + P2 as a
+reducible fiber. The answer for this question is the main theorem of this subsection:
+Theorem 5.8. Let X be a rational elliptic surface. Then X has a 1-gap if and only if r = 0 or
+r = 1 and π has a III∗ fiber.
+20
+
+Our strategy for the proof is the following. We already know that a 1-gap exists whenever r = 0
+(Theorem 3.1) or when r = 1 and π has a III∗ fiber (Theorem 5.7, No. 43). Conversely, we need to
+find P1, P2 with P1 · P2 = 1 in all cases with r ≥ 1 and T ̸= E7.
+First we introduce two lemmas, which solve most cases with little computation, and leave the
+remaining ones for the proof of Theorem 5.8. In both Lemma 5.9 and Lemma 5.11 our goal is to
+analyze the narrow lattice E(K)0 and apply Proposition 4.12 to detect cases without a 1-gap.
+Lemma 5.9. If one of the following holds, then h(P) = 4 for some P ∈ E(K)0.
+a) The Gram matrix of E(K)0 has a 4 in its main diagonal.
+b) There is an embedding of An ⊕ Am in E(K)0 for some n, m ≥ 1.
+c) There is an embedding of An, Dn or En in E(K)0 for some n ≥ 3.
+Proof. Case a) is trivial. Assuming b), we take generators P1, P2 from An, Am respectively with
+h(P1) = h(P2) = 2. Since An, Am are in direct sum, ⟨P1, P2⟩ = 0, hence h(P1 + P2) = 4, as desired.
+If c) holds, then the fact that n ≥ 3 allows us to choose two elements P1, P2 among the generators
+of L1 = An, Dn or En such that h(P1) = h(P2) = 2 and ⟨P1, P2⟩ = 0. Thus h(P1 + P2) = 4 as
+claimed. ■
+Corollary 5.10. In the following cases, X does not have a 1-gap.
+• r ≥ 3 : all cases except possibly No. 20.
+• r = 1, 2 : cases No. 25, 26, 30, 32-36, 38, 41, 42, 46, 52, 54, 60.
+Proof. We look at column E(K)0 in Table 8 to find which cases satisfy one of the conditions a),
+b), c) from Lemma 5.9.
+a) Applies to No. 12, 17, 19, 22, 23, 25, 30, 32, 33, 36, 38, 41, 46, 52, 54, 60.
+b) Applies to No. 10, 11, 14, 15, 18, 24, 26, 34, 35, 42.
+c) Applies to No. 1-10, 13, 16, 21.
+In particular, this covers all cases with r ≥ 3 (No. 1-24) except No. 20. By Lemma 5.9 in each
+of these cases there is P ∈ E(K)0 with h(P) = 4 and we are done by Proposition 4.12 1). ■
+In the next lemma we also analyze E(K)0 to detect surfaces without a 1-gap.
+Lemma 5.11. Assume E(K)0 ≃ An for some n ≥ 1 and that E(K) has nontrivial torsion part.
+Then X does not have a 1-gap. This applies to cases No. 28, 39, 44, 48, 51, 57, 58 in Table 8.
+Proof. Take a generator P of E(K)0 with h(P) = 2 and apply Proposition 4.12 2). ■
+21
+
+We are ready to prove the main result of this subsection.
+Proof of Theorem 5.8. We need to show that in all cases where r ≥ 1 and T ̸= E7 there are
+P1, P2 ∈ E(K) such that P1 · P2 = 1. This corresponds to cases No. 1-61 except 43 in Table 8.
+The cases where r = 1 and E(K) is torsion-free can be solved by Theorem 5.10, namely No.
+45-47, 49, 50, 55, 56. Adding these cases to the ones treated in Corollary 5.10 and Lemma 5.11,
+we have therefore solved the following:
+No. 1-19, 21-26, 28, 30, 32-36, 38, 39, 41-52, 54-58, 60.
+For the remaining cases, we apply Proposition 4.12 3), which involves finding perfect squares
+in the interval
+�
+4−cmax
+µ
+, 4−cmin
+µ
+�
+(see Table 7), considering the half-open interval in the cases with
+∆ = 2 (No. 53, 61).
+No.
+T
+E(K)
+µ
+I
+n2 ∈ I
+20
+A⊕2
+2
+⊕ A1
+A∗
+2 ⊕ ⟨1/6⟩
+1
+6
+[13, 23]
+42
+27
+E6
+A∗
+2
+2
+3
+[4, 4]
+22
+29
+A5 ⊕ A1
+A∗
+1 ⊕ ⟨1/6⟩
+1
+6
+[12, 21]
+42
+31
+A4 ⊕ A2
+1
+15
+�
+2
+1
+1
+8
+�
+2
+15
+[16, 21]
+42
+37
+A3 ⊕ A2 ⊕ A1
+A∗
+1 ⊕ ⟨1/12⟩
+1
+12
+[22, 28]
+52
+40
+A⊕2
+2
+⊕ A⊕2
+1
+⟨1/6⟩⊕2
+1
+6
+[10, 21]
+42
+53
+A5 ⊕ A⊕2
+1
+⟨1/6⟩ ⊕ Z/2Z
+1
+6
+[9, 12]
+32
+59
+A3 ⊕ A2 ⊕ A⊕2
+1
+⟨1/12⟩ ⊕ Z/2Z
+1
+12
+[16, 42]
+42, 52, 62
+61
+A⊕3
+2
+⊕ A1
+⟨1/6⟩ ⊕ Z/3Z
+1
+6
+[9, 12]
+32
+Table 7: Perfect squares in the interval I :=
+�
+4−cmax
+µ
+, 4−cmin
+µ
+�
+.
+In No. 59 we have ∆ > 2, so a particular treatment is needed. Let T = Tv1 ⊕ Tv2 ⊕ Tv3 ⊕ Tv4 =
+A3 ⊕ A2 ⊕ A1 ⊕ A1. If P generates the free part of E(K) and Q generates its torsion part, then
+h(P) =
+1
+12 and 4P + Q meets the reducible fibers at Θv1,2, Θv2,1, Θv3,1, Θv4,1 [Kur14][Example 1.7].
+By Table 1 and the height formula (2),
+42
+12 = h(4P + Q) = 2 + 2(4P + Q) · O − 2 · 2
+4
+− 1 · 2
+3
+− 1
+2 − 1
+2,
+hence (4P + Q) · O = 1, as desired. ■
+22
+
+6
+Appendix
+We reproduce part of the table in [OS91, Main Th.] with data on Mordell-Weil lattices of
+rational elliptic surfaces with Mordell-Weil rank r ≥ 1. We only add columns cmax, cmin, ∆.
+No.
+r
+T
+E(K)0
+E(K)
+cmax
+cmin
+∆
+1
+8
+0
+E8
+E8
+0
+0
+0
+2
+7
+A1
+E7
+E∗
+8
+1
+2
+1
+2
+0
+3
+6
+A2
+E6
+E∗
+6
+2
+3
+2
+3
+0
+4
+A⊕2
+1
+D6
+D∗
+6
+3
+2
+1
+1
+2
+5
+5
+A3
+D5
+D∗
+5
+1
+3
+4
+1
+4
+6
+A2 ⊕ A1
+A5
+A∗
+5
+7
+6
+1
+2
+2
+3
+7
+A⊕3
+1
+D4 ⊕ A1
+D∗
+4 ⊕ A∗
+1
+3
+2
+1
+2
+1
+8
+4
+A4
+A4
+A∗
+4
+6
+5
+4
+5
+2
+5
+9
+D4
+D4
+D∗
+4
+1
+1
+0
+10
+A3 ⊕ A1
+A3 ⊕ A1
+A∗
+3 ⊕ A∗
+1
+3
+2
+1
+2
+1
+11
+A⊕2
+2
+A⊕2
+2
+A∗
+2
+⊕2
+4
+3
+2
+3
+2
+3
+12
+A2 ⊕ A⊕2
+1
+
+
+
+
+
+4
+−1
+0
+1
+−1
+2
+−1
+0
+0
+−1
+2
+−1
+1
+0
+−1
+2
+
+
+
+
+
+1
+6
+
+
+
+
+
+2
+1
+0
+−1
+1
+5
+3
+1
+0
+3
+6
+3
+−1
+1
+3
+5
+
+
+
+
+
+5
+3
+1
+2
+7
+6
+13
+A⊕4
+1
+D4
+D∗
+4 ⊕ Z/2Z
+2
+1
+2
+3
+2
+14
+A⊕4
+1
+A⊕4
+1
+A∗
+1
+⊕4
+2
+1
+2
+3
+2
+15
+3
+A5
+A2 ⊕ A1
+A∗
+2 ⊕ A∗
+1
+3
+2
+5
+6
+2
+3
+16
+D5
+A3
+A∗
+3
+5
+4
+1
+1
+4
+17
+A4 ⊕ A1
+
+
+
+4
+−1
+1
+−1
+2
+−1
+1
+−1
+2
+
+
+
+1
+10
+
+
+
+3
+1
+−1
+1
+7
+3
+−1
+3
+7
+
+
+
+17
+10
+1
+2
+6
+5
+18
+D4 ⊕ A1
+A⊕3
+1
+A∗
+1
+⊕3
+3
+2
+1
+2
+1
+19
+A3 ⊕ A2
+
+
+
+2
+0
+−1
+0
+2
+−1
+−1
+−1
+4
+
+
+
+1
+12
+
+
+
+7
+1
+2
+1
+7
+2
+2
+2
+4
+
+
+
+5
+3
+2
+3
+1
+23
+
+20
+A⊕2
+2
+⊕ A1
+A2 ⊕ ⟨6⟩
+A∗
+2 ⊕ ⟨1/6⟩
+11
+6
+1
+2
+4
+3
+21
+A3 ⊕ A⊕2
+1
+A3
+A∗
+3 ⊕ Z/2Z
+2
+1
+2
+3
+2
+22
+A3 ⊕ A⊕2
+1
+A1 ⊕ ⟨4⟩
+A∗
+1 ⊕ ⟨1/4⟩
+2
+1
+2
+3
+2
+23
+A2 ⊕ A⊕3
+1
+A1 ⊕
+�
+4
+−2
+−2
+4
+�
+A∗
+1 ⊕ 1
+6
+�
+2
+1
+1
+2
+�
+13
+6
+1
+2
+5
+3
+24
+A⊕5
+1
+A⊕3
+1
+A∗
+1
+⊕3 ⊕ Z/2Z
+5
+2
+1
+2
+2
+25
+2
+A6
+�
+4
+−1
+−1
+2
+�
+1
+7
+�
+2
+1
+1
+4
+�
+12
+7
+6
+7
+6
+7
+26
+D6
+A⊕2
+1
+A∗
+1
+⊕2
+3
+2
+1
+1
+2
+27
+E6
+A2
+A∗
+2
+4
+3
+4
+3
+0
+28
+A5 ⊕ A1
+A2
+A∗
+2 ⊕ Z/2Z
+2
+1
+2
+3
+2
+29
+A5 ⊕ A1
+A1 ⊕ ⟨6⟩
+A∗
+1 ⊕ ⟨1/6⟩
+2
+1
+2
+3
+2
+30
+D5 ⊕ A1
+A1 ⊕ ⟨4⟩
+A∗
+1 ⊕ ⟨1/4⟩
+7
+4
+1
+2
+5
+4
+31
+A4 ⊕ A2
+�
+8
+−1
+−1
+2
+�
+1
+15
+�
+2
+1
+1
+8
+�
+28
+15
+2
+3
+6
+5
+32
+D4 ⊕ A2
+�
+4
+−2
+−2
+4
+�
+1
+6
+�
+2
+1
+1
+2
+�
+5
+3
+2
+3
+1
+33
+A4 ⊕ A⊕2
+1
+�
+6
+−2
+−2
+4
+�
+1
+10
+�
+2
+1
+1
+3
+�
+11
+5
+1
+2
+17
+10
+34
+D4 ⊕ A⊕2
+1
+A⊕2
+1
+A∗
+1
+⊕2
+2
+1
+2
+3
+2
+35
+A⊕2
+3
+A⊕2
+1
+A∗
+1
+⊕2 ⊕ Z/2Z
+2
+3
+4
+5
+4
+36
+A⊕2
+3
+⟨4⟩⊕2
+⟨1/4⟩⊕2
+2
+3
+4
+5
+4
+37
+A3 ⊕ A2 ⊕ A1
+A1 ⊕ ⟨12⟩
+A∗
+1 ⊕ ⟨1/12⟩
+13
+6
+1
+2
+5
+3
+38
+A3 ⊕ A⊕3
+1
+A1 ⊕ ⟨4⟩
+A∗
+1 ⊕ ⟨1/4⟩ ⊕ Z/2Z
+5
+2
+1
+2
+2
+39
+A⊕3
+2
+A2
+A∗
+2 ⊕ Z/3Z
+2
+2
+3
+4
+3
+40
+A⊕2
+2
+⊕ A⊕2
+1
+⟨6⟩⊕2
+⟨1/6⟩⊕2
+7
+3
+1
+2
+11
+6
+24
+
+41
+A2 ⊕ A⊕4
+1
+�
+4
+−2
+−2
+4
+�
+1
+6
+�
+2
+1
+1
+2
+�
+8
+3
+1
+2
+13
+6
+42
+A⊕6
+1
+A⊕2
+1
+A∗
+1
+⊕2 ⊕ (Z/2Z)2
+3
+1
+2
+5
+2
+43
+1
+E7
+A1
+A∗
+1
+3
+2
+3
+2
+0
+44
+A7
+A1
+A∗
+1 ⊕ Z/2Z
+2
+7
+8
+11
+8
+45
+A7
+⟨8⟩
+⟨1/8⟩
+2
+7
+8
+11
+8
+46
+D7
+⟨4⟩
+⟨1/4⟩
+7
+4
+1
+3
+4
+47
+A6 ⊕ A1
+⟨14⟩
+⟨1/14⟩
+31
+14
+1
+2
+12
+7
+48
+D6 ⊕ A1
+A1
+A∗
+1
+2
+3
+2
+1
+2
+49
+E6 ⊕ A1
+⟨6⟩
+⟨1/6⟩
+11
+6
+1
+2
+4
+3
+50
+D5 ⊕ A2
+⟨12⟩
+⟨1/12⟩
+23
+12
+2
+3
+5
+4
+51
+A5 ⊕ A2
+A1
+A∗
+1 ⊕ Z/3Z
+13
+6
+2
+3
+3
+2
+52
+D5 ⊕ A⊕2
+1
+⟨4⟩
+⟨1/4⟩ ⊕ Z/2Z
+9
+4
+1
+2
+7
+4
+53
+A5 ⊕ A⊕2
+1
+⟨6⟩
+⟨1/6⟩ ⊕ Z/2Z
+5
+2
+1
+2
+2
+54
+D4 ⊕ A3
+⟨4⟩
+⟨1/4⟩ ⊕ Z/2Z
+2
+3
+4
+5
+4
+55
+A4 ⊕ A3
+⟨20⟩
+⟨1/20⟩
+11
+5
+3
+4
+29
+20
+56
+A4 ⊕ A2 ⊕ A1
+⟨30⟩
+⟨1/30⟩
+71
+30
+1
+2
+28
+15
+57
+D4 ⊕ A⊕3
+1
+A1
+A∗
+1
+5
+2
+1
+2
+2
+58
+A⊕2
+3
+⊕ A1
+A1
+A∗
+1 ⊕ Z/4Z
+5
+2
+1
+2
+2
+59
+A3 ⊕ A2 ⊕ A⊕2
+1
+⟨12⟩
+⟨1/12⟩ ⊕ Z/2Z
+8
+3
+1
+2
+13
+6
+60
+A3 ⊕ A⊕4
+1
+⟨4⟩
+⟨1/4⟩ ⊕ Z/2Z
+3
+1
+2
+5
+2
+61
+A⊕3
+2
+⊕ A1
+⟨6⟩
+⟨1/6⟩ ⊕ Z/3Z
+5
+2
+1
+2
+2
+Table 8:
+Mordell-Weil lattices of rational elliptic surfaces
+with Mordell-Weil rank r ≥ 1.
+25
+
+References
+[Ber12] P. Bernays. Über die Darstellung von positiven, ganzen Zahlen durch die primitive, binären
+quadratischen Formen einer nicht-quadratischen Diskriminante. PhD thesis, Göttingen,
+1912.
+[BH]
+M. Bhargava and J. Hanke. Universal quadratic forms and the 290-Theorem. Preprint at
+http://math.stanford.edu/~vakil/files/290-Theorem-preprint.pdf.
+[Cos]
+R. D. Costa.
+Classification of fibers of conic bundles on rational elliptic surfaces.
+arXiv:2206.03549.
+[Elk90]
+N. D. Elkies. The Mordell-Weil lattice of a rational elliptic surface. Arbeitstagung Bonn,
+1990.
+[HW79] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Clarendon
+Press, 1979.
+[Kur14] Y. Kurumadani. Pencil of cubic curves and rational elliptic surfaces.
+Master’s thesis,
+Kyoto University, 2014.
+[MP89] R. Miranda and U. Persson. Torsion groups of elliptic surfaces. Compositio Mathematica,
+72(3):249–267, 1989.
+[Nis96]
+K. Nishiyama. The Jacobian fibrations on some K3 surfaces and their Mordell-Weil groups.
+Japanese Journal of Mathematics, 22(2), 1996.
+[OS91]
+K. Oguiso and T. Shioda. The Mordell-Weil lattice of a rational elliptic surface. Com-
+mentarii Mathematici Universitatis Sancti Pauli, 40, 1991.
+[Shi89]
+T. Shioda. The Mordell-Weil lattice and Galois representation, I, II, III. Proceedings of
+the Japan Academy, 65(7), 1989.
+[Shi90]
+T. Shioda. On the Mordell-Weil lattices. Commentarii Mathematici Universitatis Sancti
+Pauli, 39(7), 1990.
+[SS10]
+M. Schuett and T. Shioda.
+Elliptic surfaces.
+Advanced Studies in Pure Mathematics,
+60:51–160, 2010.
+[SS19]
+M. Schuett and T. Shioda. Mordell-Weil Lattices, volume 70 of Ergebnisse der Mathematik
+und ihrer Grenzgebiete. Springer, 2019.
+26
+

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@@ -0,0 +1,2290 @@
+COMPUTING NONSURJECTIVE PRIMES ASSOCIATED TO GALOIS
+REPRESENTATIONS OF GENUS 2 CURVES
+BARINDER S. BANWAIT, ARMAND BRUMER, HYUN JONG KIM, ZEV KLAGSBRUN, JACOB MAYLE,
+PADMAVATHI SRINIVASAN, AND ISABEL VOGT
+Abstract. For a genus 2 curve C over Q whose Jacobian A admits only trivial geometric en-
+domorphisms, Serre’s open image theorem for abelian surfaces asserts that there are only finitely
+many primes ℓ for which the Galois action on ℓ-torsion points of A is not maximal. Building on
+work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are
+Mitchell’s classification of maximal subgroups of PSp4(Fℓ), sampling of the characteristic polyno-
+mials of Frobenius, and the Khare–Wintenberger modularity theorem. The algorithm has been
+submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomor-
+phism ring in the LMFDB, and the results incorporated into the homepage of each such curve on
+a publicly-accessible branch of the LMFDB.
+1. Introduction
+Let C/Q be a smooth, projective, geometrically integral curve (referred to hereafter as a nice
+curve) of genus 2, and let A be its Jacobian. We assume throughout that A admits no nontrivial
+geometric endomorphisms; that is, we assume that End(AQ) = Z, and we refer to any abelian
+variety satisfying this property as typical1. We also say that a nice curve is typical if its Jacobian is
+typical. Let GQ ∶= Gal(Q/Q), let ℓ be a prime, and let A[ℓ] ∶= A(Q)[ℓ] denote the ℓ-torsion points
+of A(Q). Let
+ρA,ℓ ∶ GQ → Aut(A[ℓ])
+denote the Galois representation on A[ℓ].
+By fixing a basis for A[ℓ], and observing that A[ℓ]
+admits a nondegenerate Galois-equivariant alternating bilinear form, namely the Weil pairing, we
+may identify the codomain of ρA,ℓ with the general symplectic group GSp4(Fℓ).
+In a letter to Vign´eras [Ser00, Corollaire au Th´eor`eme 3], Serre proved an open image theorem
+for typical abelian varieties of dimensions 2 or 6, or of odd dimension, generalizing his celebrated
+open image theorem for elliptic curves [Ser72]. More precisely, the set of nonsurjective primes ℓ for
+which the representation ρA,ℓ is not surjective — i.e., the set of primes ℓ for which ρA,ℓ(GQ) is
+contained in a proper subgroup of GSp4(Fℓ) — is finite.
+In the elliptic curve case, Serre subsequently provided a conditional upper bound in terms of the
+conductor of E on this finite set [Ser81, Th´eor`eme 22]; this bound has since been made unconditional
+[Coj05, Kra95]. There are also algorithms to compute the finite set of nonsurjective primes [Zyw15],
+and practical implementations in Sage [CL12].
+Serre’s open image theorem for typical abelian surfaces was made explicit by Dieulefait [Die02]
+who described an algorithm that returns a finite set of primes containing the set of nonsurjective
+primes. In a different direction Lombardo [Lom16, Theorem 1.3] provided an upper bound on the
+nonsurjective primes involving the stable Faltings height of A.
+Date: January 6, 2023.
+2010 Mathematics Subject Classification. 11F80 (primary), 11G10, 11Y16 (secondary).
+1Abelian varieties with extra endomorphisms define a thin set (in the sense of Serre) in Ag and as such are not
+the typically arising case.
+1
+arXiv:2301.02222v1  [math.NT]  5 Jan 2023
+
+2
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+In this paper we develop Algorithms 3.1 and 4.1, which together allow for the exact determination
+of the nonsurjective primes for C, yielding our main result as follows.
+Theorem 1.1. Let C/Q be a typical genus 2 curve whose Jacobian A has conductor N.
+(1) Algorithm 3.1 produces a finite list PossiblyNonsurjectivePrimes(C) that provably contains all
+nonsurjective primes.
+(2) For a given bound B > 0, Algorithm 4.1 produces a sublist LikelyNonsurjectivePrimes(C;B)
+of PossiblyNonsurjectivePrimes(C) that contains all the nonsurjective primes.
+If B is suffi-
+ciently large, then the elements of LikelyNonsurjectivePrimes(C;B) are precisely the nonsurjec-
+tive primes of A.
+The two common ingredients in Algorithms 3.1 and 4.1 are Mitchell’s 1914 classification of
+maximal subgroups of PSp4(Fℓ) [Mit14] and sampling of characteristic polynomials of Frobenius
+elements. Indeed, ρA,ℓ is nonsurjective precisely when its image is contained in one of the proper
+maximal subgroups of GSp4(Fℓ). The (integral) characteristic polynomial of Frobenius at a good
+prime p is computationally accessible since it is determined by counting points on C over Fpr for
+small r. The reduction of this polynomial modulo ℓ gives the characteristic polynomial of the action
+of the Frobenius element on A[ℓ]. By the Chebotarev density theorem, the images of the Frobenius
+elements for varying primes p equidistribute over the conjugacy classes of ρA,ℓ(GQ) and hence let
+us explore the image.
+Algorithm 3.1 makes use of the fact that if the image of ρA,ℓ is nonsurjective, then the character-
+istic polynomials of Frobenius at auxiliary primes p will be constrained modulo ℓ. Using this idea,
+Dieulefait worked out the constraints imposed by each type of maximal subgroup for ρA,ℓ(GQ) to
+be contained in that subgroup. Our Algorithm 3.1 combines Dieulefait’s conditions, with some
+modest improvements, to produce a finite list PossiblyNonsurjectivePrimes(C).
+Algorithm 4.1 then weeds out the extraneous surjective primes from PossiblyNonsurjectivePrimes(C).
+Equipped with the prime ℓ, the task here is try to generate enough different elements in the image
+to rule out containment in any proper maximal subgroup. The key input is a purely group-theoretic
+condition (Proposition 4.2) that guarantees that a subgroup is all of GSp4(Fℓ) if it contains par-
+ticular types of elements. This algorithm is probabilistic and depends on the choice of a parameter
+B which, if sufficiently large, provably establishes nonsurjectivity. The parameter B is a cut-off for
+the number of Frobenius elements that we use to sample the conjugacy classes of ρA,ℓ(GQ).
+As an illustration of the interplay between theory and practice, analyzing the “worst case” run
+time of each step in Algorithm 3.1 yields a new theoretical bound, conditional on the Generalized
+Riemann Hypothesis (GRH), on the product of all nonsurjective primes in terms of the conductor.
+Theorem 1.2. Let C/Q be a typical genus 2 curve with conductor N. Assuming the Generalized
+Riemann Hypothesis (GRH), we have, for any ϵ > 0,
+∏
+ℓ nonsurjective
+ℓ ≪ exp(N1/2+ϵ),
+where the implied constant is absolute and effectively computable.
+While we believe this bound to be far from asymptotically optimal, it is the first bound in the
+literature expressed in terms of the (effectively computable) conductor.
+Naturally one wants to find the sufficiently large value of B in Theorem 1.1(2), which the next
+result gives, conditional on GRH.
+Theorem 1.3. Let C/Q be a typical genus 2 curve, B be a positive integer, and q be the largest
+prime in LikelyNonsurjectivePrimes(C;B). Assuming GRH, the set LikelyNonsurjectivePrimes(C;B)
+is precisely the set of nonsurjective primes of C, provided that
+B ≥ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5)
+2 .
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+3
+The proof of Theorem 1.3 involves an explicit Chebotarev bound due to Bach and Sorenson
+[BS96] that is dependent on GRH. An unconditional version of Theorem 1.3 can be given using an
+unconditional Chebotarev result (for instance [KW22]), though the bound for B will be exponential
+in q. In addition, if we assume both GRH and the Artin Holomorphy Conjecture (AHC), then a
+version of Theorem 1.3 holds with the improved asymptotic bound B ≫ q11 log2(qNA), but without
+an explicit constant.
+Unfortunately, the bound from Theorem 1.3 is prohibitively large to use in practice. By way of
+illustration, consider the smallest (with respect to conductor) typical genus 2 curve, which has a
+model
+y2 + (x3 + 1)y = x2 + x,
+and label 249.a.249.1 in the L-functions and modular forms database (LMFDB) [LMF22]. The
+output of Algorithm 3.1 is the set {2,3,5,7,83}. Applying Algorithm 4.1 with B = 100 rules out
+the prime 83, suggesting that 7 is the largest nonsurjective prime. Subsequently applying Theorem
+1.3 with q = 7 yields the value B = 3.578 × 1023 for which LikelyNonsurjectivePrimes(C;B) coincides
+with the set of nonsurjective primes associated with C. With this value of B, our implementation of
+the algorithm was still running after 24 hours, after which we terminated it. Even if the version of
+Theorem 1.3 that relies on AHC could be made explicit, the value of q11 log2(qNA) in this example
+is on the order of 1011, which would still be a daunting prospect.
+To execute the combined algorithm on all typical genus 2 curves in the LMFDB - which at the
+time of writing constitutes 63,107 curves - we have decided to take a fixed value of B = 1000 in
+Algorithm 4.1. The combined algorithm then takes about 4 hours on MIT’s Lovelace computer,
+a machine with 2 AMD EPYC 7713 2GHz processors, each with 64 cores, and a total of 2TB of
+memory. The result of this computation of nonsurjective primes for these curves is available to
+view on the homepage of each curve in the LMFDB beta:
+https://beta.lmfdb.org
+In addition, the combined algorithm has been run on a much larger set of 1,823,592 curves
+provided to us by Andrew Sutherland. See Section 6 for the results of this computation.
+Algorithm 4.1 samples the characteristic polynomial of Frobenius Pp(t) for each prime p of
+good reduction for the curve up to a particular bound and applies Tests 4.4 and 4.5 to Pp(t).
+Assuming that ρA,ℓ is surjective, we expect that the outcome of these tests should be independent
+for sufficiently large primes. More precisely,
+Theorem 1.4. Let C/Q be a typical genus 2 curve with Jacobian A and suppose ℓ is an odd prime
+such that ρA,ℓ is surjective. There is an effective bound B0 such that for any B > B0, if we sample
+the characteristic polynomials of Frobenius Pp(t) for n primes p ∈ [B,2B] chosen uniformly and
+independently at random, the probability that none of these pass Tests 4.4 or 4.5 is less than 3⋅( 9
+10)
+n.
+Remark 1. In fact, for each prime ℓ satisfying the conditions of Theorem 1.4, there is an explicit
+constant cℓ ≤
+9
+10 tending to 3
+4 as ℓ → ∞ which may be computed using Corollary 5.3 such that
+bound of 3 ⋅ ( 9
+10)
+n in Theorem 1.4 can be replaced by 3 ⋅ cn
+ℓ .
+The combined algorithm to probabilistically determine the nonsurjective primes of a nice genus
+2 curve over Q has been implemented in Sage [The20], and it will appear in a future release of this
+software2. Until then, the implementation is available at the following repository:
+https://github.com/ivogt/abeliansurfaces
+The README.md file contains detailed instructions on its use. This repository also contains other
+scripts in both Sage and Magma [BCP97] useful for verifying some of the results of this work; any
+filenames used in the sequel will refer to the above repository.
+2see https://trac.sagemath.org/ticket/30837 for the ticket tracking this integration.
+
+4
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+Outline of this paper. In Section 2, we begin by reviewing the properties of the characteristic
+polynomial of Frobenius with a view towards computational aspects. We also recall the classification
+of maximal subgroups of GSp4(Fℓ). In Section 3, we explain Algorithm 3.1 and establish Theorem
+1.1(1); that is, for each of the maximal subgroups of GSp4(Fℓ) listed in Section 2.4, we generate a
+list of primes that provably contains all primes ℓ for which the mod ℓ image of Galois is contained
+in this maximal subgroup. Theorem 1.2 is also proved in this section (Subsection 3.3). In Section 4,
+we first prove a group-theoretic criterion (Proposition 4.2) for a subgroup of GSp4(Fℓ) to equal
+GSp4(Fℓ). Then, for each ℓ in the finite list from Section 3, we ascertain whether the characteristic
+polynomials of the Frobenius elements sampled satisfy the group-theoretic criterion; Theorem 1.1(2)
+and Theorem 1.3 also follow from this study. In Section 5 we prove Theorem 1.4 concerning the
+probability of output error, assuming that Frobenius elements distribute in ρA,ℓ(GQ) as they would
+in a randomly chosen element of GSp4(Fℓ). Finally, in Section 6, we close with remarks concerning
+the execution of the algorithm on the large dataset of genus 2 curves mentioned above, and highlight
+some interesting examples that arose therein.
+Acknowledgements. This work was started at a workshop held remotely ‘at’ the Institute for
+Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, in May
+2020, and was supported by a grant from the Simons Foundation (546235) for the collaboration
+‘Arithmetic Geometry, Number Theory, and Computation’.
+It has also been supported by the
+National Science Foundation under Grant No. DMS-1929284 while the authors were in residence
+at ICERM during a Collaborate@ICERM project held in May 2022. We are grateful to Noam Elkies
+for providing interesting examples of genus 2 curves in the literature, Davide Lombardo for helpful
+discussions related to computing geometric endomorphism rings, and to Andrew Sutherland for
+providing a dataset of Hecke characteristic polynomials that were used for executing our algorithm
+on all typical genus 2 curves in the LMFDB, as well as making available the larger dataset of
+approximately 2 million curves that we ran our algorithm on.
+2. Preliminaries
+2.1. Notation. Let A be an abelian variety of dimension g defined over Q. By conductor we mean
+the Artin conductor N = NA of A. We write Nsq for the largest integer such that N2
+sq ∣ N.
+Let ℓ be a prime. We write TℓA for the ℓ-adic Tate module of A:
+TℓA ≃ lim
+←�
+n
+A[ℓn].
+This is a free Zℓ-module of rank 2g.
+For each prime p, we write Frobp ∈ Gal(Q/Q) for an absolute Frobenius element associated to p.
+By a good prime p for an abelian variety A, we mean a prime p for which A has good reduction, or
+equivalently p ∤ NA. If p is a good prime for A, then the trace ap of the action of Frobp on TℓA is
+an integer. See Section 2.2 for a discussion of the characteristic polynomial of Frobenius.
+By a typical abelian variety A, we mean an abelian variety with geometric endomorphism ring
+Z. A typical genus 2 curve is a nice curve whose Jacobian is a typical abelian surface.
+Let V be a 4-dimensional vector space over Fℓ endowed with a nondegenerate skew-symmetric
+bilinear form ⟨⋅,⋅⟩. A subspace W ⊆ V is called isotropic (for ⟨⋅,⋅⟩) if ⟨w1,w2⟩ = 0 for all w1,w2 ∈ W.
+A subspace W ⊆ V is called nondegenerate (for ⟨⋅,⋅⟩) if ⟨⋅,⋅⟩ restricts to a nondegenerate form on
+W. The general symplectic group of (V,⟨⋅,⋅⟩) is defined as
+GSp(V,⟨⋅,⋅⟩) ∶= {M ∈ GL(V ) ∶ ∃ mult(M) ∈ F×
+ℓ ∶ ⟨Mv,Mw⟩ = mult(M)⟨v,w⟩ ∀ v,w ∈ V }.
+The map M ↦ mult(M) is a surjective homomorphism from GSp(V,⟨⋅,⋅⟩) to F×
+ℓ called the similitude
+character; its kernel is the symplectic group, denoted Sp(V,⟨⋅,⋅⟩).
+Usually the bilinear form is
+understood from the context, in which case one drops ⟨⋅,⋅⟩ from the notation; moreover, for our
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+5
+purposes, we will have fixed a basis for V , one in which the bilinear form is represented by the
+nonsingular skew-symmetric matrix
+J ∶= ( 0
+I2
+−I2
+0 ),
+where I2 is the 2 × 2 identity matrix.
+By a subquotient W of a Galois module U, we mean a Galois module W that admits a surjection
+U ′ ↠ W from a subrepresentation U ′ of U.
+Since we are chiefly concerned with computing the sets LikelyNonsurjectivePrimes(C;B) and
+PossiblyNonsurjectivePrimes(C) for a fixed curve C, we will henceforth, for ease of notation, drop
+the C from the notation for these sets.
+2.2. Integral characteristic polynomial of Frobenius. The theoretical result underlying the
+whole approach is the following.
+Theorem 2.1 (Weil, see [ST68, Theorem 3]). Let A be an abelian variety of dimension g defined
+over Q and let p be a prime of good reduction for A. Then there exists a monic integral polynomial
+Pp(t) ∈ Z[t] of degree 2g with constant coefficient pg such that for any ℓ ≠ p, the polynomial Pp(t)
+modulo ℓ is the characteristic polynomial of the action of Frobp on TℓA. Furthermore, every root
+of Pp(t) has complex absolute value p1/2.
+The polynomials Pp(t) are computationally accessible by counting points on C over Fpr r = 1,2.
+See [Poo17, Chapter 7] for more details.
+In fact, Pp(t) can be accessed via the frobenius_
+polynomial command in Sage. In particular, we denote the trace of Frobenius by ap. By the
+Grothendieck-Lefschetz trace formula, if A = JacX, p is a prime of good reduction for X, and
+λ1,...,λ2g are the roots of Pp(t), then
+#X(Fpr) = pr + 1 −
+2g
+∑
+i=1
+λr
+i .
+2.3. The Weil pairing and consequences on the characteristic polynomial of Frobenius.
+The nondegenerate Weil pairing gives an isomorphism (of Galois modules):
+(1)
+TℓA ≃ (TℓA)∨ ⊗Zℓ Zℓ(1).
+The Galois character acting on Zℓ(1) is the ℓ-adic cyclotomic character, which we denote by cycℓ.
+The integral characteristic polynomial for the action of Frobp on Zℓ(1) is simply t−p. The integral
+characteristic polynomial for the action of Frobp on (TℓA)∨ is the reversed polynomial
+P ∨
+p (t) = Pp(1/t) ⋅ t2g/pg
+whose roots are the inverses of the roots of Pp(t).
+We now record a few easily verifiable consequences of the nondegeneracy of the Weil pairing
+when dim(A) = 2.
+Lemma 2.2.
+(i) The roots of Pp(t) come in pairs that multiply out to p. In particular, Pp(t) has no root with
+multiplicity 3.
+(ii) Pp(t) = t4 − apt3 + bpt2 − papt + p2 for some ap,bp ∈ Z.
+(iii) If the trace of an element of GSp4(Fℓ) is 0 mod ℓ, then its characteristic polynomial is re-
+ducible modulo ℓ. In particular, this applies to Pp(t) when ap ≡ 0 (mod ℓ).
+(iv) If A[ℓ] is a reducible GQ-module, then Pp(t) is reducible modulo ℓ.
+Proof. Parts (i) and (ii) are immediate from the fact that the non-degenerate Weil pairing allows
+us to pair up the four roots of Pp(t) into two pairs that each multiply out to p.
+
+6
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+For part (iii), suppose that M ∈ GSp4(Fℓ) has tr(M) = 0. Then the characteristic polynomial
+PM(t) of M is of the form t4 +bt2 +c2. When the discriminant of PM is 0 modulo ℓ, the polynomial
+PM has repeated roots and is hence reducible. So assume that the discriminant of PM is nonzero
+modulo ℓ. When ℓ ≠ 2, the result follows from [Car56, Theorem 1]. When ℓ = 2, a direct computation
+shows that the characteristic polynomial of a trace 0 element of GSp4(F2) is either (t + 1)4 or
+(t2 + t + 1)2, which are both reducible.
+Part (iv) is immediate from Theorem 2.1 since Pp(t) mod ℓ by definition is the characteristic
+polynomial for the action of Frobp on A[ℓ].
+□
+2.4. Maximal subgroups of GSp4(Fℓ). Mitchell [Mit14] classified the maximal subgroups of
+PSp4(Fℓ) in 1914. This can be used to deduce the following classification of maximal subgroups of
+GSp4(Fℓ) with surjective similitude character.
+Lemma 2.3 (Mitchell). Let V be a 4-dimensional Fℓ-vector space endowed with a nondegener-
+ate skew-symmetric bilinear form ω. Then any proper subgroup G of GSp(V,ω) with surjective
+similitude character is contained in one of the following types of maximal subgroups.
+(1) Reducible maximal subgroups
+(a) Stabilizer of a 1-dimensional isotropic subspace for ω.
+(b) Stabilizer of a 2-dimensional isotropic subspace for ω.
+(2) Irreducible subgroups governed by a quadratic character
+Normalizer Gℓ of the group Mℓ that preserves each summand in a direct sum decomposition
+V1 ⊕ V2 of V , where V1 and V2 are jointly defined over Fℓ and either:
+(a) both nondegenerate for ω; or
+(b) both isotropic for ω.
+Moreover, Mℓ is an index 2 subgroup of Gℓ.
+(3) Stabilizer of a twisted cubic
+GL(W) acting on Sym3 W ≃ V , where W is a 2-dimensional Fℓ-vector space.
+(4) Exceptional subgroups See Table A for explicit generators for the groups described below.
+(a) When ℓ ≡ ±3 (mod 8): a group whose image G1920 in PGSp(V,ω) has order 1920.
+(b) When ℓ ≡ ±5 (mod 12) and ℓ ≠ 7: a group whose image G720 in PGSp(V,ω) has order 720.
+(c) When ℓ = 7: a group whose image G5040 in PGSp(V,ω) has order 5040.
+Remark 2. We have chosen to label the maximal subgroups in the classification using invariant
+subspaces for the symplectic pairing ω on V , following the more modern account due to Aschbacher
+(see [Lom16, Section 3.1]; for a more comprehensive treatment see [KL90]). For the convenience of
+the reader, we record the correspondence between Mitchell’s original labels and ours below.
+Mitchell’s label
+Label in Lemma 2.3
+Group having an invariant point and plane
+1a
+Group having an invariant parabolic congruence
+1b
+Group having an invariant hyperbolic or elliptic congruence
+2a
+Group having an invariant quadric
+2b
+Table 1. Dictionary between maximal subgroup labels in [Die02]/[Mit14] and Lemma 2.3
+Remark 3. The maximal subgroups in (1) are the analogues of the Borel subgroup of GL2(Fℓ).
+The maximal subgroups in (2) when the two subspaces V,V ′ in the direct sum decomposition
+are individually defined over Fℓ are the analogues of normalizers of the split Cartan subgroup of
+GL2(Fℓ). When the two subspaces V,V ′ are not individually defined over Fℓ instead, the maximal
+subgroups in (2) are analogues of the normalizers of the non-split Cartan subgroups of GL2(Fℓ).
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+7
+Remark 4. We briefly explain why the action of GL2(Fℓ) on Sym3(F2
+ℓ) preserves a nondegenerate
+symplectic form. It suffices to show that the restriction to SL2(Fℓ) fixes a vector in ⋀2 Sym3(F2
+ℓ).
+This follows by character theory. If W is the standard 2-dimensional representation of SL2, then
+we have ⋀2(Sym3 W) ≃ Sym4 W ⊕ 1 as representations of SL2.
+Remark 5. One can extract explicit generators of the exceptional maximal subgroups from Mitchell’s
+original work3. Indeed [Mit14, the proof of Theorem 8, page 390] gives four explicit matrices that
+generate a G1920 (which is unique up to conjugacy in PGSp4(Fℓ)). Mitchell’s description of the
+other exceptional groups is in terms of certain projective linear transformations called skew perspec-
+tivities attached to a direct sum decomposition V = V1 ⊕ V2 into 2-dimensional subspaces. A skew
+perspectivity of order n with axes V1 and V2 is the projective linear transformation that scales V1 by
+a primitive nth root of unity and fixes V2. This proof also gives the axes of the skew perspectivities
+of order 2 and 3 that generate the remaining exceptional groups [Mit14, pages 390-391]. Table 5
+lists generators of (one representative of the conjugacy class of) each of the exceptional maximal
+subgroup extracted from Mitchell’s descriptions. In the file exceptional.m publicly available
+with our code, we verify that Magma’s list of conjugacy classes of maximal subgroups of GSp4(Fℓ)
+agree with those described in Lemma 2.3 for 3 ≤ ℓ ≤ 47.
+Remark 6. The classification of exceptional maximal subgroups of PSp4(Fℓ) is more subtle than
+that of PGSp4(Fℓ), because of the constraint on the similitude character of matrices in PSp4(Fℓ).
+While the similitude character is not well-defined on PGSp4(Fℓ) (multiplication by a scalar c ∈ F×
+ℓ
+scales the similitude character by c2) it is well-defined modulo squares. The group PSp4(Fℓ) is the
+kernel of this natural map:
+1 → PSp4(Fℓ) → PGSp4(Fℓ)
+mult
+��→ F×
+ℓ /(F×
+ℓ )2 ≃ {±1} → 1.
+An exceptional subgroup of PGSp4(Fℓ) gives rise to an exceptional subgroup of PSp4(Fℓ) of either
+the same size or half the size depending on the image of mult restricted to that subgroup, which
+in turn depends on the congruence class of ℓ. For this reason, the maximal exceptional subgroups
+of PSp4(Fℓ) in Mitchell’s original classification (also recalled in Dieulefait [Die02, Section 2.1]) can
+have order 1920 or 960 and 720 or 360 depending on the congruence class of ℓ, and 2520 (for
+ℓ = 7). Such an exceptional subgroup gives rise to a maximal exceptional subgroup of PGSp4(Fℓ)
+only when mult is surjective (i.e., its intersection with PSp4(Fℓ) is index 2), which explains the
+restricted congruence classes of ℓ for which they arise.
+We now record a lemma that directly follows from the structure of maximal subgroups described
+above. This lemma will be used in Section 4 to devise a criterion for a subgroup of GSp4(Fℓ) to be
+the entire group. For an element T in GSp4(Fℓ), let tr(T), mid(T), mult(T) denote the trace of
+T, the middle coefficient of the characteristic polynomial of T, and the similitude character applied
+to T respectively4. For a scalar λ, we have
+tr(λT) = λtr(T),
+mid(λT) = λ2 mid(T),
+mult(λT) = λ2 mult(T).
+Hence the quantities tr(T)2/mult(T) and mid(T)/mult(T) are well-defined on PGSp4(Fℓ). For
+ℓ > 2 and ∗ ∈ {720,1920,5040}, define
+(2)
+Cℓ,∗ ∶= {( tr(T)2
+mult(T), mid(T)
+mult(T)) ∣ T ∈ an exceptional subgroup of GSp4(Fℓ) of projective order ∗}
+Lemma 2.4.
+(1) In cases 2a and 2b of Lemma 2.3:
+3Mitchell’s notation for PGSp4(Fℓ) is Aν(ℓ) and for PSp4(Fℓ) is A1(ℓ).
+4Explicitly, the characteristic polynomial of T is therefore t4 − tr(T)t3 + mid(T)t2 − mult(T) tr(T)t + mult(T)2.
+
+8
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+(a) every element in Gℓ ∖ Mℓ has trace 0, and,
+(b) the group Mℓ stabilizes a non-trivial linear subspace of F
+4
+ℓ.
+(2) Every element that is contained in a maximal subgroup corresponding to the stabilizer of a
+twisted cubic has a reducible characteristic polynomial.
+(3) For ∗ ∈ {1920,720}, the set Cℓ,∗ defined in (2) equals the reduction modulo ℓ of the elements of
+the set C∗ below.
+C1920 = {(0,−2),(0,−1),(0,0),(0,1),(0,2),(1,1),(2,1),(2,2),(4,2),(4,3),(8,4),(16,6)}
+C720 = {(0,1),(0,0),(4,3),(1,1),(16,6),(0,2),(1,0),(3,2),(0,−2)}
+We also have
+C7,5040 = {(0,0),(0,1),(0,2),(0,5),(0,6),(1,0),(1,1),(2,6),(3,2),(4,3),(5,3),(6,3)}.
+Proof.
+(1) In cases 2a and 2b of Lemma 2.3, since any element of the normalizer Gℓ that is not in Mℓ
+switches elements in the two subspaces V1 and V2 (i.e. maps elements in the subspace V1
+in the decomposition V1 ⊕ V2 to elements in V2 and vice-versa), it follows that any element
+in Gℓ ∖ Mℓ has trace zero.
+(2) The conjugacy class of maximal subgroups corresponding to the stabilizer of a twisted cubic
+comes from the embedding GL2(Fℓ)
+ι�→ GSp4(Fℓ) induced by the natural action of GL2(Fℓ)
+on the space of monomials of degree 3 in 2 variables. If M is a matrix in GL2(Fℓ) with
+eigenvalues λ,µ (possibly repeated), then the eigenvalues of ι(M) are λ3,µ3,λ2µ,λµ2 and
+hence the characteristic polynomial of ι(M) factors as (T 2 −(λ3 +µ3)T +λ3µ3)(T 2 −(λ2µ+
+λµ2)T + λ3µ3) over Fℓ which is reducible over Fℓ.
+(3) This follows from the description of the maximal subgroups given in Table 5. Each case
+(except G5040 that only occurs for ℓ = 7) depends on a choice of a root of a quadratic
+polynomial. In the file exceptional statistics.sage, we generate the corresponding
+finite subgroups over the appropriate quadratic number field to compute C∗. It follows that
+the corresponding values for the subgroup G∗ in GSp4(Fℓ) can be obtained by reducing
+these values modulo ℓ. Since the group G5040 only appears for ℓ = 7, we directly compute
+the set C7,5040.
+□
+Remark 7. The condition in Lemma 2.4(3) is the analogue of the condition [Ser72, Proposition 19
+(iii)] used to rule out exceptional maximal subgroups of GL2(Fℓ).
+We end this subsection by including the following lemma, to further highlight the similarities
+between the above classification of maximal subgroups of GSp4(Fℓ) and the more familiar classi-
+fication of maximal subgroups of GL2(Fℓ). This lemma is not used elsewhere in the article and is
+thus for expositional purposes only.
+Lemma 2.5.
+(1) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are indi-
+vidually defined over Fℓ is isomorphic to
+{(m1,m2) ∈ GL2(Fℓ)2 ∣ det(m1) = det(m2)}.
+In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ2 − 1)2.
+(2) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are individually
+defined over Fℓ is isomorphic to
+{(m1,m2) ∈ GL2(Fℓ)2 ∣ mT
+1 m2 = λI, for some λ ∈ F∗
+ℓ }.
+In particular, the order of Mℓ is ℓ(ℓ − 1)2(ℓ2 − 1).
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+9
+(3) The subgroup Mℓ in the case (2a) when the two nondegenerate subspaces V1 and V2 are not
+individually defined over Fℓ is isomorphic to
+{m ∈ GL2(Fℓ2) ∣ det(m) ∈ F∗
+ℓ }.
+In particular, the order of Mℓ is ℓ2(ℓ − 1)(ℓ4 − 1).
+(4) The subgroup Mℓ in the case (2b) when the two isotropic subspaces V1 and V2 are not indi-
+vidually defined over Fℓ is isomorphic to GU2(Fℓ2), i.e.,
+{m ∈ GL2(Fℓ2) ∣ mT ι(m) = λI, for some λ ∈ F∗
+ℓ },
+where ι denotes the natural extension of the Galois automorphism of Fℓ2/Fℓ to GL2(Fℓ2). In
+particular, the order of Mℓ is ℓ(ℓ2 − 1)2.
+Proof. Given a direct sum decomposition V1 ⊕ V2 of a vector space V over Fq, we get a natural
+embedding of Aut(V1) × Aut(V2) (≅ GL2(Fq)2) into Aut(V ) (≅ GL4(Fq)), whose image consists of
+automorphisms that preserve this direct sum decomposition. We will henceforth refer to elements
+of Aut(V1) × Aut(V2) as elements of Aut(V ) using this embedding. To understand the subgroup
+Mℓ of GSp4(Fq) in cases (1) and (2) where the two subspaces in the direct sum decomposition are
+individually defined over Fq, we need to further impose the condition that the automorphisms in
+the image of the map Aut(V1) × Aut(V2) → Aut(V ) preserve the symplectic form ω on V up to a
+scalar.
+In (1), without any loss of generality, the two nondegenerate subspaces V1 and V2 can be chosen
+to be orthogonal complements under the nondegenerate pairing ω, and so by Witt’s theorem, in a
+suitable basis for V1⊕V2 obtained by concatenating a basis of V1 and a basis of V2, the nondegenerate
+symplectic pairing ω has the following block-diagonal shape:
+B ∶=
+⎡⎢⎢⎢⎢⎢⎢⎢⎣
+0
+1
+−1
+0
+0
+1
+−1
+0
+⎤⎥⎥⎥⎥⎥⎥⎥⎦
+.
+The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing
+up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which boils down to
+det(m1) = λ = det(m2).
+Similarly, in (2), without any loss of generality, by Witt’s theorem, in a suitable basis for V1 ⊕V2
+obtained by concatenating a basis of the isotropic subspace V1 and a basis of the isotropic subspace
+V2, the nondegenerate symplectic pairing ω has the following block-diagonal shape.
+B ∶=
+⎡⎢⎢⎢⎢⎢⎢⎢⎣
+0
+1
+1
+0
+0
+−1
+−1
+0
+⎤⎥⎥⎥⎥⎥⎥⎥⎦
+.
+The condition that an element (m1,m2) ∈ Aut(V1) ⊕ Aut(V2) preserves the symplectic pairing
+up to a similitude factor of λ is the condition (m1,m2)T B(m1,m2) = λB, which again boils down
+to mT
+1 m2 = λI.
+If we have a subspace W defined over Fq2 but not defined over Fq, and we let W denote the
+conjugate subspace and further assume that W ⊕W gives a direct sum decomposition of V , then we
+get a natural embedding of Aut(W) (≅ GL2(Fq2)) into Aut(V ) (≅ GL4(Fq)) whose image consists
+of automorphisms that commute with the natural involution of V ⊗ Fq2 induced by the Galois
+automorphism of Fq2 over Fq. The proofs of cases (3) and (4) are analogous to the cases (1) and (2)
+respectively, by using the direct sum decomposition W ⊕W and letting m2 = ι(m1). The condition
+that det(m1) = det(m2) in (1) becomes the condition det(m1) = det(m2) = detm1 = det(m1), or
+
+10
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+equivalently, that det(m1) ∈ Fq in (3). Similarly, the condition that mT
+1 m2 = λI in (2) becomes the
+condition that mT
+1 ι(m1) = λI in (4).
+□
+2.5. Image of inertia and (tame) fundamental characters. Dieulefait [Die02] used Mitchell’s
+work described in the previous subsection to classify the maximal subgroups of GSp4(Fℓ) that could
+occur as the image of ρA,ℓ . This was achieved via an application of a fundamental result of Serre
+and Raynaud that strongly constrains the action of inertia at ℓ, and which we now recall.
+Fix a prime ℓ > 3 that does not divide the conductor N of A. Let Iℓ be an inertia subgroup
+at ℓ. Let ψn∶Iℓ → F×
+ℓn denote a (tame) fundamental character of level n. The n Galois-conjugate
+fundamental characters ψn,1,...,ψn,n of level n are given by ψn,i ∶= ψℓi
+n . Recall that the fundamental
+character of level 1 is simply the mod ℓ cyclotomic character cycℓ, and that the product of all
+fundamental characters of a given level is the cyclotomic character.
+Theorem 2.6 (Serre [Ser72], Raynaud [Ray74], cf. [Die02][Theorem 2.1). Let ℓ be a semistable
+prime for A. Let V /Fℓ be an n-dimensional Jordan–H¨older factor of the Iℓ-module A[ℓ]. Then V
+admits a 1-dimensional Fℓn-vector space structure such that ρA,ℓ∣Iℓ acts on V via the character
+ψd1
+n,1⋯ψdn
+n,n
+with each di equal to either 0 or 1.
+On the other hand, the following fundamental result of Grothendieck constrains the action of
+inertia at semistable primes p ≠ ℓ.
+Theorem 2.7 (Grothendieck [GRR72, Expos´e IX, Prop 3.5]). Let A be an abelian variety over a
+number field K. Then A has semistable reduction at p ≠ ℓ if and only if the action of Ip ⊂ GK on
+TℓA is unipotent of length 2.
+Combining these two results allows one fine control of the determinant of a subquotient of A[ℓ];
+this will be used in Section 3.
+Corollary 2.8. Let A/Q be an abelian surface, and let Xℓ be a Jordan–H¨older factor of the Fℓ[GQ]-
+module A[ℓ] ⊗ Fℓ. If ℓ is a semistable prime, then
+detXℓ ≃ ϵ ⋅ cycx
+ℓ
+for some character ϵ∶GQ → Fℓ that is unramified at ℓ and some 0 ≤ x ≤ dimXℓ. Moreover, ϵ120 = 1.
+Proof. The first part follows immediately from Theorem 2.6.
+For the fact that ϵ120 = 1, every
+abelian surface attains semistable reduction over an extension K/Q with [K ∶ Q] dividing 120 by
+[LV14a, Theorem 7.2], and so this follows from Theorem 2.7 since there are no nontrivial unramified
+characters of GQ.
+□
+We can now state Dieulefait’s classification of maximal subgroups of GSp4(Fℓ) that can occur
+as the image ρA,ℓ(GQ) for a semistable prime ℓ > 7.
+Proposition 2.9 ([Die02]). Let A be the Jacobian of a genus 2 curve defined over Q with Weil
+pairing ω on A[ℓ]. If ℓ > 7 is a semistable prime, then ρA,ℓ(GQ) is either all of GSp(A[ℓ],ω) or it
+is contained in one of the maximal subgroups of Types (1) or (2) in Lemma 2.3.
+See also [Lom16, Proposition 3.15] for an expanded exposition of why the image of GQ cannot
+be contained in maximal subgroup of Type (3) for a semistable prime ℓ > 7.
+Remark 8. However, if ℓ is a prime of additive reduction, or if ℓ ≤ 7, then the image of GQ may also
+be contained in any of the four types of maximal subgroups described in Lemma 2.3. Nevertheless,
+by [LV22, Theorem 6.6], for any prime ℓ > 24, we have that the exponent of the projective image is
+bounded exp(PρA,ℓ) ≥ (ℓ−1)/12. Since exp(G1920) = 2exp(S6) = 120 and exp(G720) = exp(S5) = 60,
+the exceptional maximal subgroups cannot occur as ρA,ℓ(GQ) for ℓ > 1441.
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+11
+2.6. A consequence of the Chebotarev density theorem. Let K/Q be a finite Galois exten-
+sion with Galois group G = Gal(K/Q) and absolute discriminant dK. Let S ⊆ G be a nonempty
+subset that is closed under conjugation. By the Chebotarev density theorem, we know that
+(3)
+lim
+x→∞
+∣{p ≤ x ∶ p is unramified in K and Frobp ∈ S}∣
+∣{p ≤ x}∣
+= ∣S∣
+∣G∣.
+Let p be the least prime such that p is unramified in K and Frobp ∈ S. There are effective versions
+of the Chebotarev density theorem that give bounds on p. The best known unconditional bounds
+are polynomial in dK [LMO79, AK19, KW22]. Under GRH, the best known bounds are polynomial
+in log dK. In particular Bach and Sorenson [BS96] showed that under GRH,
+(4)
+p ≤ (4log dK + 2.5[K ∶ Q] + 5)2.
+The present goal is to give an effective version of the Chebotarev density theorem in the context
+of abelian surfaces. We will use a corollary of (4) that is noted in [MW21] which allows for the
+avoidance of a prescribed set of primes by taking a quadratic extension of K. We do this because
+we will take K = Q(A[ℓ]), and p being unramified in K is not sufficient to imply that p is a prime
+of good reduction for A. Lastly, we will use that by [Ser81, Proposition 6], if K/Q is finite Galois,
+then
+(5)
+log dK ≤ ([K ∶ Q] − 1)log rad(dK) + [K ∶ Q]log([K ∶ Q]),
+where radn = ∏p∣n p denotes the radical of an integer n.
+Lemma 2.10. Let A/Q be a typical principally polarized abelian surface with conductor NA. Let q
+be a prime. Let S ⊆ ρA,q(GQ) be a nonempty subset that is closed under conjugation. Let p be the
+least prime of good reduction for A such that p ≠ q and ρA,q(Frobp) ∈ S. Assuming GRH, we have
+p ≤ (4[(2q11 − 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5)
+2 .
+Proof. Let K = Q(A[q]). Then K/Q is Galois and
+[K ∶ Q] ≤ ∣GSp4(Fq)∣ = q4(q4 − 1)(q2 − 1)(q − 1) ≤ q11.
+As raddK is the product of primes that ramify in Q(A[q]), the criterion of N´eron-Ogg-Shafarevich
+for abelian varieties [ST68, Theorem 1] implies that rad(dK) divides rad(qNA). Let ˜K ∶= K(√m)
+where m ∶= rad(2NA). Note that the primes that ramify in ˜K are precisely 2, q, and the primes of
+bad reduction for A. Thus rad(d ˜
+K) = rad(2qNA). Moreover [ ˜K ∶ Q] ≤ 2q11 and by (5),
+log(d ˜
+K) ≤ (2q11 − 1)log rad(2qNA) + 22q11 log(2q).
+Applying [MW21, Corollary 6] to the field ˜K, we get that (under GRH) there exists a prime p
+satisfying the claimed bound, that does not divide m, and for which ρA,q(Frobp) ∈ S.
+□
+3. Finding a finite set containing all nonsurjective primes
+In this section we describe Algorithm 3.1 referenced in Theorem 1.1(1). This algorithm produces
+a finite list PossiblyNonsurjectivePrimes that provably includes all nonsurjective primes ℓ. We also
+prove Theorem 1.2.
+Since our goal is to produce a finite list (from which we will later remove extraneous primes) it
+is harmless to include the finitely many bad primes as well as 2,3,5,7. Using Proposition 2.9, it
+suffices to find conditions on ℓ > 7 for which ρA,ℓ(GQ) could be contained in one of the maximal
+subgroups of type (1) and (2) in Lemma 2.3. We first find primes ℓ for which ρA,ℓ has (geometrically)
+reducible image (and hence is contained in a maximal subgroup in case (1) of Lemma 2.3 or in a
+subgroup Mℓ in case (2)). To treat the geometrically irreducible cases, we then make use of the
+observation from Lemma 2.4 1a that every element outside of an index 2 subgroup has trace 0.
+
+12
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+Algorithm 3.1. Given a typical genus 2 curve C/Q with conductor N and Jacobian A, compute
+a finite list PossiblyNonsurjectivePrimes of primes as follows.
+(1) Initialize PossiblyNonsurjectivePrimes = [2,3,5,7].
+(2) Add to PossiblyNonsurjectivePrimes all primes dividing N.
+(3) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗ Fℓ could be reducible via
+Algorithms 3.3, 3.6, and 3.10.
+(4) Add to PossiblyNonsurjectivePrimes the good primes ℓ for which ρA,ℓ ⊗Fℓ could be irreducible but
+nonsurjective via Algorithm 3.13.
+(5) Return PossiblyNonsurjectivePrimes.
+At a very high-level, each of the subalgorithms of Algorithm 3.1 makes use of a set of auxiliary
+good primes p. We compute the integral characteristic polynomial of Frobenius Pp(t) and use it to
+constrain those ℓ ≠ p for which the image could have a particular shape.
+Remark 9. Even though robust methods to compute the conductor N of a genus 2 curve are not
+implemented at the time of writing, the odd-part Nodd of N can be computed via genus2red
+function of PARI and the genus2reduction module of SageMath, both based on an algorithm
+of Liu [Liu94]. Moreover, [BK94, Theorem 6.2] bounds the 2-exponent of N above by 20 and hence
+N can be bounded above by 220Nodd. While these algorithms can be run only with the bound
+220Nodd, it will substantially increase the run-time of the limiting Algorithm 3.10.
+We now explain each of these steps in detail.
+3.1. Good primes that are not geometrically irreducible. In this section we describe the
+conditions that ℓ must satisfy for the base-extension A[ℓ] ∶= A[ℓ] ⊗Fℓ Fℓ to be reducible. In this
+case, the representation A[ℓ] is an extension
+(6)
+0 → Xℓ → A[ℓ] → Yℓ → 0
+of a (quotient) representation Yℓ by a (sub) representation Xℓ. Recall that Nsq denotes the largest
+square divisor of N.
+Lemma 3.2. Let ℓ be a prime of good reduction for A and suppose that A[ℓ] sits in sequence (6).
+Let p ≠ ℓ be a good prime for A and let f denote the order of p in (Z/NsqZ)×. Then there exists
+0 ≤ x ≤ dimXℓ and 0 ≤ y ≤ dimYℓ such that Frobgcd(f,120)
+p
+acts on detXℓ by pgcd(f,120)x, respectively
+on detYℓ by pgcd(f,120)y.
+Proof. Since ℓ is a good prime and Xℓ is composed of Jordan–H¨older factors of A[ℓ], Corollary 2.8
+constrains its determinant. We have detXℓ = ϵcycx
+ℓ for some character ϵ∶GQ → Fℓ unramified at ℓ,
+and 0 ≤ x ≤ dimXℓ, and ϵ120 = 1. Hence Frob120
+p
+acts on detXℓ by cycℓ(Frobp)120x = p120x.
+In fact, we can do slightly better. Since detA[ℓ] ≃ cyc2
+ℓ, we have detYℓ ≃ ϵ−1 cyc2−x
+ℓ
+. Since the
+conductor is multiplicative in extensions, we conclude that cond(ϵ)2 ∣ N. By class field theory,
+the character ϵ factors through (Z/cond(ϵ)Z)×, and hence through (Z/NsqZ)×, sending Frobp
+to p (mod Nsq). Since pf ≡ 1 (mod Nsq), we have that ϵ(Frobp)gcd(f,120) = 1, and we see that
+Frobgcd(f,120)
+p
+acts on detXℓ by pgcd(f,120)x. Exchanging the roles of Xℓ and Yℓ, we deduce the
+analogous statement for Yℓ.
+□
+This is often enough information to find all ℓ for which A[ℓ] has a nontrivial subquotient. Namely,
+by Theorem 2.1, every root of Pp(t) has complex absolute value p1/2. Thus the gcd(f,120)-th power
+of each root has complex absolute value pgcd(f,120)/2, and hence is never integrally equal to 1 or
+pgcd(f,120). Since Lemma 3.2 guarantees that this equality must hold modulo ℓ for any good prime
+ℓ for which A[ℓ] is reducible with a 1-dimensional subquotient, we always get a nontrivial condition
+on ℓ. Some care must be taken to rule out ℓ for which A[ℓ] only has 2-dimensional subquotient(s).
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+13
+3.1.1. Odd-dimensional subquotient. Let p be a good prime.
+Given a polynomial P(t) and an
+integer f, write P (f)(t) for the polynomial whose roots are the fth powers of roots of P(t).
+Universal formulas for such polynomials in terms of the coefficients of P(t) are easy to compute,
+and are implemented in our code in the case where P is a degree 4 polynomial whose roots multiply
+in pairs to pα, and f ∣ 120.
+Algorithm 3.3. Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of
+p in (Z/NsqZ)× and write f′ = gcd(f,120). Compute an integer Modd as follows.
+(1) Choose a nonempty finite set T of auxiliary good primes p ∤ N.
+(2) For each p, compute
+Rp ∶= P (f′)
+p
+(1).
+(3) Let Modd = gcdp∈T (pRp) over all auxiliary primes.
+Return the list of prime divisors ℓ of Modd.
+Proposition 3.4. Any good prime ℓ for which A[ℓ] has an odd-dimensional subrepresentation is
+returned by Algorithm 3.3.
+Proof. Since A[ℓ] is 4-dimensional and has an odd-dimensional subrepresentation, it has a 1-
+dimensional subquotient. For any p ∈ T , Lemma 3.2 shows that Frobf′
+p acts on detXℓ by either pf′
+or by 1. Thus, the action of Frobf′
+p on A[ℓ] has an eigenvalue that is congruent to pf′ or 1 modulo
+ℓ, and so P (f′)
+p
+(t) has a root that is congruent to 1 or pf′ modulo ℓ. Since the roots of P (f′)(t)
+multiply in pairs to pf′, we have P (f′)
+p
+(pf′) = p2f′P (f′)
+p
+(1). Hence ℓ divides p ⋅ P (f′)
+p
+(1) = pRp.
+□
+Using Theorem 2.1, we can give a theoretical bound on the “worst case” of this step of the
+algorithm using only one auxiliary prime p. Of course, taking the greatest common divisor over
+multiple auxiliary primes will likely remove extraneous factors, and in practice this step of the
+algorithm runs substantially faster than other steps.
+Proposition 3.5. Algorithm 3.3 terminates. More precisely, if p is any good prime for A, then
+0 ≠ ∣Modd∣ ≪ p240
+where the implied constant is absolute.
+Proof. This follows from the fact that the coefficient of ti in P (f′)
+p
+(t) has magnitude on the order
+of p(2−i)f′ and f′ ≤ 120.
+□
+3.1.2. Two-dimensional subquotients. We now assume that A[ℓ] is reducible, but does not have
+any odd-dimensional subquotients.
+In particular, it has an irreducible subrepresentation Xℓ of
+dimension 2, with irreducible quotient Yℓ of dimension 2. If A[ℓ] is reducible but indecomposable,
+then Xℓ is the unique subrepresentation of A[ℓ] and Y ∨
+ℓ ⊗ cycℓ is the unique subrepresentation
+of A[ℓ]
+∨ ⊗ cycℓ. The isomorphism TℓA ≃ (TℓA)∨ ⊗ cycℓ from (1) yields an isomorphism A[ℓ] ≃
+(A[ℓ])∨ ⊗ cycℓ and hence Xℓ ≃ Y ∨
+ℓ ⊗ cycℓ. Otherwise, A[ℓ] ≃ Xℓ ⊕ Yℓ and so the nondegeneracy of
+the Weil pairing gives
+Xℓ ⊕ Yℓ ≃ (X∨
+ℓ ⊗ cycℓ) ⊕ (Y ∨
+ℓ ⊗ cycℓ).
+Therefore either:
+(a) Xℓ ≃ Y ∨
+ℓ ⊗ cycℓ and Yℓ ≃ X∨
+ℓ ⊗ cycℓ, or
+(b) Xℓ ≃ X∨
+ℓ ⊗ cycℓ and Yℓ ≃ Y ∨
+ℓ ⊗ cycℓ and A[ℓ] ≃ Xℓ ⊕ Yℓ.
+We call the first case related 2-dimensional subquotients and the second case self-dual 2-dimensional
+subrepresentations.
+We will see that the ideas of Lemma 3.2 easily extend to treat the related
+subquotient case; we will use the validity of Serre’s conjecture to treat the self-dual case. In the
+
+14
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+case that A[ℓ] is decomposable, the above two cases correspond respectively to the index 2 subgroup
+Mℓ in cases (2a) (the isotropic case) and (2b) (the nondegenerate case) of Lemma 2.3.
+3.1.3. Related two-dimensional subquotients. Let p be a good prime. Let Pp(t) ∶= t4−at3+bt2−pat+p2
+be the characteristic polynomial of Frobp acting on A[ℓ]. Suppose that α and β are the eigenvalues
+of Frobp acting on the subrepresentation Xℓ. Then, since Xℓ ≃ Y ∨
+ℓ ⊗ cycℓ, the eigenvalues of the
+action of Frobp on Yℓ are p/α and p/β. The action of Frobp on detXℓ is therefore by a product of
+two of the roots of Pp(t) that do not multiply to p. Note that there are four such pairs of roots of
+Pp(t) that do not multiply to p. Let Qp(t) be the quartic polynomial whose roots are the products
+of pairs of roots of Pp(t) that do not multiply to p. By design, the roots of Qp(t) have complex
+absolute value p, but are not equal to p. (It is elementary to work out that
+Qp(t) = t4 − (b − 2p)t3 + p(a2 − 2b + 2p)t2 − p2(b − 2p)t + p4
+and is a quartic whose roots multiply in pairs to p2.)
+Algorithm 3.6. Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of
+p in (Z/NsqZ)× and write f′ = gcd(f,120). Compute an integer Mrelated as follows.
+(1) Choose a finite set T of auxiliary good primes p ∤ N;
+(2) For each p, compute the product
+Rp ∶= Q(f′)
+p
+(1)Q(f′)
+p
+(pf′)
+(3) Let Mrelated = gcdp∈T (pRp).
+Return the list of prime divisors ℓ of Mrelated.
+Proposition 3.7. Any good prime ℓ for which A[ℓ] has related two-dimensional subquotients is
+returned by Algorithm 3.6.
+Proof. Proceed similarly as in the proof of Proposition 3.4 — in particular, ℓ divides Q(f′)
+p
+(1),
+Q(f′)
+p
+(pf′), or Q(f′)
+p
+(p2f′) and hence ℓ divides pRp since Q(f′)
+p
+(p2f′) = p4f′Q(f′)
+p
+(1).
+□
+A theoretical “worst case” analysis yields the following.
+Proposition 3.8. Algorithm 3.6 terminates. More precisely, if q is the smallest surjective prime
+for A, then a good prime p for which Rp is nonzero is bounded by a function of q. Assuming GRH,
+p ≪ q22 log2(qN),
+where the implied constants are absolute and effectively computable. Moreover, for such a prime p,
+∣Mrelated∣ ≪ p961 ≪ q21142 log1922(qN),
+where the implied constants are absolute.
+Proof. By Serre’s open image theorem for genus 2 curves, such a prime q exists, and by Lemma
+2.10, the prime p can be chosen such that Rp is nonzero modulo q. Finally,
+Mrelated ≤ pRp = pQ(f′)(1)Q(f′)(pf′) ≪ p8f′+1 ≪ p961,
+since the coefficient of ti in Q(f′)(t) has magnitude on the order of p(4−i)f′ and f′ ≤ 120.
+□
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+15
+3.1.4. Self-dual two-dimensional subrepresentations. In this case, both subrepresentations Xℓ and
+Yℓ are absolutely irreducible 2-dimensional Galois representations with determinant the cyclotomic
+character cycℓ. It follows that the representations are odd (i.e., the determinant of complex con-
+jugation is −1.) Therefore, by the Khare–Wintenberger theorem (formerly Serre’s conjecture on
+the modularity of mod-ℓ Galois representations) [Kha06, KW09a, KW09b], both Xℓ and Yℓ are
+modular; that is, for i = 1,2, there exist newforms fi ∈ Snew
+ki (Γ1(Ni),ϵi) such that
+Xℓ ≅ ρf1,ℓ and Yℓ ≅ ρf2,ℓ.
+Furthermore, by the multiplicativity of Artin conductors, we obtain the divisibility N1N2 ∣ N.
+Lemma 3.9. Both f1 and f2 have weight two and trivial Nebentypus; that is, k1 = k2 = 2, and
+ϵ1 = ϵ2 = 1.
+Proof. From Theorem 2.6, we have that Xℓ∣Iℓ and Yℓ∣Iℓ must each be conjugate to either of the
+following subgroups of GL2(Fℓ):
+(1
+∗
+0
+cycℓ
+) or (ψ2
+0
+0
+ψℓ
+2
+).
+The assertion of weight 2 now follows from [Ser87, Proposition 3]. (Alternatively, one may use
+Proposition 4 of loc. cit., observing that X��� and Yℓ are finite and flat as group schemes over Zℓ
+because ℓ is a prime of good reduction.)
+From Section 1 of loc. cit., the Nebentypus ϵi of fi satisfies, for all p ∤ ℓN,
+detXℓ(Frobp) = p ⋅ ϵi(p),
+where this equality is viewed inside F
+×
+ℓ . The triviality follows.
+□
+We therefore have newforms fi ∈ Snew
+2
+(Γ0(Ni)) such that
+(7)
+A[ℓ] ≃ ρf1,ℓ ⊕ ρf2,ℓ.
+We may assume without loss of generality that N1 ≤
+√
+N. Let p ∤ N be an auxiliary prime. We
+obtain from equation (7) that the integral characteristic polynomial of Frobenius factors:
+Pp(t) ≡ (t2 − ap(f1)t + p)(t2 − ap(f2)t + p)
+mod ℓ;
+here we use the standard property that, for f a normalised eigenform with trivial Nebentypus,
+ρf,ℓ(Frobp) satisfies the polynomial equation t2 − ap(f)t + p for p ≠ ℓ. In particular, we have
+Res(Pp(t),t2 − ap(f1)t + p) ≡ 0
+mod ℓ.
+This serves as the basis of the algorithm to find all primes ℓ in this case.
+Algorithm 3.10. Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer
+Mself-dual as follows.
+(1) Compute the set S of divisors d of N with d ≤
+√
+N.
+(2) For each d ∈ S:
+(a) compute the Hecke L-polynomial
+Qd(t) ∶= ∏
+f
+(t2 − ap(f)t + p),
+where the product is taken over the finitely many newforms in Snew
+2
+(Γ0(d));
+(b) choose a finite set T of auxiliary primes p ∤ N;
+(c) for each auxiliary prime p, compute the resultant
+Rp(d) ∶= Res(Pp(t),Qd(t));
+
+16
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+(d) Take the greatest common divisor
+M(d) ∶= gcd
+p∈T
+(pRp(d)).
+(3) Let Mself-dual ∶= ∏d∈S M(d).
+Return the list of prime divisors ℓ of Mself-dual.
+Proposition 3.11. Any good prime ℓ for which A[ℓ] has self-dual two-dimensional subrepresenta-
+tions is returned by Algorithm 3.10.
+Proof. If ℓ is in T for any d ∈ S, then ℓ is in the output because Mself-dual is a multiple of M(d)
+which in turn is a multiple of any element of T . Otherwise, as explained before Algorithm 3.10,
+there is some N1 ∈ S and some newform f1 ∈ Snew
+2
+(Γ0(N1)) such that Res(Pp(t),t2 − apf1t + p) ≡ 0
+(mod ℓ) for every p ∈ T . In particular, Rp(N1) ≡ 0 (mod ℓ), so ℓ divides M(N1) and Mself-dual.
+□
+We can again do a “worst case” theoretical analysis of this algorithm to conclude the following.
+As this indicates, this is by far the limiting step of the algorithm.
+Proposition 3.12. Algorithm 3.10 terminates. More precisely, if q is the smallest surjective prime
+for A, then a good prime p for which Rp(d) is nonzero is bounded by a function of q. Assuming GRH,
+p ≪ q22 log2(qN), where the implied constant is absolute and effectively computable. Moreover, for
+such a prime p, we have
+∣Rp(d)∣ ≪ (2p1/2)8 dim Snew
+2
+(Γ0(d)) ≪ (4p)(d+1)/3,
+and so all together
+∣Mself-dual∣ ≪ (4q)N1/2+ϵ,
+where the implied constants are absolute.
+Proof. As in Proposition 3.8, we use Serre’s open image theorem and the Effective Chebotarev
+Theorem. If Rp(d) is zero integrally, then in particular Rp(d) ≡ 0 (mod q) and Pp(t) is reducible
+modulo q. Since GSp4(Fq) contains elements that do not have reducible characteristic polynomial,
+Lemma 2.10 implies that such elements are the image of Frobp for p bounded as claimed.
+The resultant Rp(d) is the product of the pairwise differences of the roots of Pp(t) and Qd(t),
+which all have complex absolute value p1/2. Hence the pairwise differences have absolute value
+at most 2p1/2.
+Moreover dimSnew
+2
+(Γ0(d)) ≤ (d + 1)/12 by [Mar05, Theorem 2].
+Since there
+are 8dimSnew
+2
+(Γ0(d)) such terms multiplied to give Rp(d), the bound for Rp(d) follows. Since
+Mself-dual = ∏ d∣N
+d≤
+√
+N
+pRp(d), it suffices to bound
+∑
+d∣N
+d≤
+√
+N
+d + 4
+3
+≤
+∑
+d∣N
+d≤
+√
+N
+√
+N + 4
+3
+≤ σ0(N)
+√
+N + 4
+3
+.
+Since σ0(N) ≪ Nϵ by [Apo76, (31) on page 296], we obtain the claimed bound.
+□
+Remark 10. The polynomial Qd(t) in step (2) of Algorithm 3.10 is closely related to the charac-
+teristic polynomial Hd(t) of the Hecke operator Tp acting on the space S2(Γ0(d)), which may be
+computed via modular symbols computations. One may recover Qd(t) from Hd(t) by first homoge-
+nizing H with an auxiliary variable z (say) to obtain Hd(t,z), and setting t = 1+pz2 (an observation
+we made in conjunction with Joseph Wetherell). In our computation of nonsurjective primes for
+the database of genus 2 curves with conductor at most 220 (including those in the LMFDB), we
+only needed to use polynomials Qd(t) for level up to 210 (since step (1) of the Algorithm has a
+√
+N term). We are grateful to Andrew Sutherland for providing us with a precomputed dataset
+for these levels resulting from the creation of an extensive database of modular forms going well
+beyond what was previously available [BBB+21].
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+17
+Remark 11. Our Sage implementation uses two auxiliary primes in Step 2(b) of the above algorithm.
+Increasing the number of such primes yields smaller supersets at the expense of longer runtime.
+3.2. Good primes that are geometrically irreducible. Let φ be any quadratic Dirichlet char-
+acter φ∶(Z/NZ)× → {±1}. Our goal in this subsection is to find all good primes ℓ governed by φ,
+by which we mean that
+tr(ρA,ℓ(Frobp)) ≡ ap ≡ 0
+mod ℓ
+whenever φ(p) = −1.
+We will consider the set of all quadratic Dirichlet character φ∶(Z/NZ)× → {±1}. Using the struc-
+ture theorem for finite abelian groups and the fact that φ factors through (Z/NZ)×/((Z/NZ)×)2,
+this set has the structure of an F2-vector space of dimension
+d(N) ∶= ω(N) +
+⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
+0
+∶ v2(N) = 0
+−1
+∶ v2(N) = 1
+0
+∶ v2(N) = 2
+1
+∶ v2(N) ≥ 3,
+where ω(m) denotes the number of prime factors of m and v2(m) is the 2-adic valuation of m. In
+particular, d(N) ≤ ω(N) + 1.
+Algorithm 3.13. Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer Mquad
+as follows.
+(1) Compute the set S of quadratic Dirichlet characters φ∶(Z/NZ)× → {±1}.
+(2) For each φ ∈ S:
+(a) Choose a nonempty finite set T of “auxiliary” primes p ∤ N for which ap ≠ 0 and φ(p) = −1.
+(b) Take the greatest common divisor
+Mφ ∶= gcd
+p∈T
+(pap),
+over all auxiliary primes p.
+(3) Let Mquad ∶= ∏φ∈S Mφ.
+Return the list of prime divisors ℓ of Mquad.
+Proposition 3.14. Any good prime ℓ for which A[ℓ] is governed by a quadratic character is
+returned by Algorithm 3.13.
+Proof. Suppose that A[ℓ] is governed by the quadratic character φ∶(Z/NZ)× → {±1}. Then for
+every good prime p ≠ ℓ for which φ(p) = −1, the prime ℓ must divide the integral trace of Frobenius
+ap. Hence ℓ divides Mφ and Mquad.
+□
+Proposition 3.15. Algorithm 3.13 terminates. More precisely, if q is the smallest surjective prime
+for A, then a good prime p for which φ(p) = −1 and ap is nonzero is bounded by a function of q.
+Assuming GRH, p ≪ 22d(N)q22 log2(qN), where the implied constant is absolute and effectively
+computable. Moreover, we have
+∏
+φ∈S
+∏
+ℓ governed
+by φ
+ℓ ≪ (23d(N)q33 log3(qN))2−21−d(N) ≪ 26ω(N)q66 log6(qN),
+where the implied constant is absolute and effectively computable.
+Proof. We imitate the proof of [LV14b, Lemma 21] in our setting. Let V be the d-dimensional
+F2-vector space of quadratic Dirichlet characters of modulus N (equivalently, quadratic Galois
+characters unramified outside of N). Let ρV ∶GK → V ∨ denote the representation sending Frobp to
+the linear functional φ ↦ φ(p). Since the character for PGSp4(Fq)/PSp4(Fq) is the abelianization
+
+18
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+of PρA,q, we conclude in the same way as [LV14b, Proof of Lemma 21] that for any α ∈ V ∨, there
+exists an Xα ∈ GSp4(Fq) with tr(Xα) ≠ 0 such that (α,Xα) is in the image of ρV × ρA,ℓ.
+Apply the effective Chebotarev density theorem to the Galois extension corresponding to ρV ×
+ρA,q. This has degree at most 2d(N)∣GSp4(Fq)∣ and is unramified outside of qN. Therefore, assum-
+ing GRH and combining (4) and (5), there exists a prime
+pα ≪ 22d(N)q22 log2(qN)
+for which (α,Xα) = (ρV (Frobpα),ρA,q(Frobpα)). Let φ be a character not in the kernel of α. Any
+exceptional prime ℓ governed by φ must divide pαapα, which is nonzero because it is nonzero modulo
+q. This proves that the algorithm terminates, since every φ is not in the kernel of precisely half
+of all α ∈ V ∨. We now bound the size of the product of all ℓ governed by a character in S. If ℓ is
+governed by φ, then ℓ divides the quantity
+p∣ap∣ ≤ p3/2 ≪ 23d(N)q33 log3(qN).
+Taking the product over all nonzero α in V (of which there are 2d(N) − 1), each ℓ will show up half
+the time, so we obtain:
+⎛
+⎜⎜⎜
+⎝
+∏
+ℓ governed
+by φ ∈ S
+ℓ
+⎞
+⎟⎟⎟
+⎠
+2d(N)−1
+≪ (23d(N)q33 log3(qN))
+2d(N)−1
+,
+which implies the result by taking the (2d(N)−1)th root of both sides.
+□
+Putting all of these pieces together, we obtain the following.
+Proof of Theorem 1.1(1). If ρA,ℓ is nonsurjective, ℓ > 7, and ℓ ∤ N, then Proposition 2.9 implies
+that ρA,ℓ(GQ) must be in one of the maximal subgroups of Type (1) or (2) listed in Lemma
+2.3. If it is contained in one of the reducible subgroups, i.e. the subgroups of Type (1), then
+ρA,ℓ(GQ) (and, hence, ρA,ℓ(GQ) ⊗ Fℓ) is reducible, and so ℓ is added to PossiblyNonsurjectivePrimes
+in Step (3) by Propositions 3.4, 3.7, and 3.11.
+If ρA,ℓ(GQ) is contained in one of the index 2
+subgroups Mℓ of an irreducible subgroup of Type (2) listed in Lemma 2.3, then again ℓ is added to
+PossiblyNonsurjectivePrimes in Step (3), since Mℓ ⊗ Fℓ is always reducible by Lemma 2.4(1b).
+Hence we may assume that ρA,ℓ(GQ) is contained in one of the irreducible maximal subgroups
+Gℓ of Type (2) listed in Lemma 2.3, but not in the index 2 subgroup Mℓ. The normalizer character
+GQ
+ρA,ℓ
+��→ Gℓ → Gℓ/Mℓ = {±1}
+is nontrivial and unramified outside of N, and so it corresponds to a quadratic Dirichlet character
+φ∶(Z/NZ)× → {±1}. Lemma 2.4(1a) shows that tr(g) = 0 in Fℓ for any g ∈ Gℓ ∖ Mℓ. Consequently,
+ℓ is governed by φ (in the language of Section 3.2), so ℓ is added to PossiblyNonsurjectivePrimes in
+Step (4) by Proposition 3.14.
+□
+3.3. Bounds on Serre’s open image theorem. In this section we combine the theoretical worst
+case bounds in the Algorithms 3.3, 3.6, 3.10, and 3.13 to give a bound on the smallest surjective
+good prime q, and the product of all nonsurjective primes, thereby establishing Theorem 1.2.
+Corollary 3.16. Let A/Q be a typical genus 2 Jacobian of conductor N. Assuming GRH, we have
+∏
+ℓ nonsurjective
+ℓ ≪ exp(N1/2+ϵ),
+where the implied constant is absolute and effectively computable.
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+19
+Proof. Let q be the smallest surjective good prime for A, which is finite by Serre’s open image
+theorem. Multiplying the bounds in Propositions 3.5, 3.8, 3.12, and 3.15 by the conductor N, the
+product of all nonsurjective primes is bounded by a function of q and N of the following shape
+(8)
+∏
+ℓ nonsurjective
+ℓ ≪ qN1/2+ϵ.
+On the other hand, since q is the smallest surjective prime by definition, the product of all primes
+less than q divides the product of all nonsurjective primes. Using [Ser81, Lemme 11], we have
+exp(q) ≪ ∏
+ℓ<q
+ℓ ≤
+∏
+ℓ nonsurjective
+ℓ ≪ qN1/2+ϵ.
+Combining the first and last terms, we have q ≪ N1/2+ϵ log(q), whence q ≪ N1/2+ϵ. Plugging this
+back into (8) yields the claimed bound.
+□
+4. Testing surjectivity of ρA,ℓ
+In this section we establish Theorem 1.1(2). The goal is to weed out any extraneous nonsur-
+jective primes in the output PossiblyNonsurjectivePrimes of Algorithm 3.1 to produce a smaller list
+LikelyNonsurjectivePrimes(B) containing all nonsurjective primes (depending on a chosen bound
+B) by testing the characteristic polynomials of Frobenius elements up to the bound B. If B is
+sufficiently large (quantified in Section 5), the list LikelyNonsurjectivePrimes(B) is provably the list
+of nonsurjective primes.
+Algorithm 4.1. Given an integer B and the output PossiblyNonsurjectivePrimes of Algorithm 3.1
+run on the typical hyperelliptic genus 2 curve with equation y2 + h(x)y = f(x), output a sublist
+LikelyNonsurjectivePrimes(B) of PossiblyNonsurjectivePrimes as follows.
+(1) Initialize LikelyNonsurjectivePrimes(B) as PossiblyNonsurjectivePrimes.
+(2) Remove 2 from LikelyNonsurjectivePrimes(B) if the size of the Galois group of the splitting field
+of 4f + h2 is 720.
+(3) For each good prime p < B, while LikelyNonsurjectivePrimes(B) is nonempty:
+(a) Compute the integral characteristic polynomial Pp(t) of Frobp.
+(b) For each prime ℓ in LikelyNonsurjectivePrimes(B), run Tests 4.4(i), (ii), and (iii) on Pp(t)
+to rule out ρA,ℓ(GQ) being contained in one of the exceptional maximal subgroups.
+(c) For each prime ℓ in LikelyNonsurjectivePrimes(B), run Tests 4.5(i) and (ii) on Pp(t) to rule
+out ρA,ℓ(GQ) being contained in one of the nonexceptional maximal subgroups.
+(d) For a given prime ℓ, if each of the 5 tests Tests 4.4(i)–(iii) and Tests 4.5(i)–(ii) have
+succeeded for some prime p, remove ℓ from LikelyNonsurjectivePrimes(B).
+(4) Return LikelyNonsurjectivePrimes(B).
+Remark 12. In our implementation of Step 3 of this algorithm, we have chosen to only use primes
+p of good reduction for the curve as auxiliary primes, which is a stronger condition than being a
+good prime for the Jacobian A. More precisely, the primes that are good for the Jacobian but bad
+for the curve are precisely the prime factors of the discriminant 4f + h2 of a minimal equation for
+the curve that do not divide the conductor NA of the Jacobian. At such a prime, the reduction
+of the curve consists of two elliptic curves E1 and E2 intersecting transversally at a single point.
+Since there are many auxiliary primes p < B to choose from, excluding bad primes for the curve is
+not a serious restriction, but allows us to access the characteristic polynomial of Frobenius directly
+by counting points on the reduction of the curve. This is not strictly necessary: one could use the
+characteristic polynomials of Frobenius for the elliptic curves E1 and E2, which can be computed
+using the genus2reduction module of SageMath.
+We briefly summarize the contents of this section. In Section 4.1, we first prove a purely group-
+theoretic criterion for a subgroup of GSp4(Fℓ) to equal the whole group. Then in Section 4.2,
+
+20
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+we explain Test 4.4 and Test 4.5, whose validity follows immediately from Lemma 2.4(3) and
+Proposition 4.2 respectively. The main idea of these tests is to use auxiliary good primes p ≠ ℓ to
+generate characteristic polynomials in the image of ρA,ℓ. If we find enough types of characteristic
+polynomials to rule out each proper maximal subgroup of GSp4(Fℓ) (cf. Proposition 4.2), then we
+can conclude that ρA,ℓ is surjective. In Section 4.3, we prove Theorems 1.1(2) and 1.3 that justify
+this algorithm.
+4.1. A group-theoretic criterion. We now use the classification of maximal subgroups of GSp4(Fℓ)
+described in Section 2.4 to deduce a group-theoretic criterion for a subgroup G of GSp4(Fℓ) to be
+the whole group. This is analogous to [Ser72, Proposition 19 (i)-(ii)].
+Proposition 4.2. Fix a prime ℓ ≠ 2 and a subgroup G ⊆ GSp4(Fℓ) with surjective similitude
+character. Assume that G is not contained in one of the exceptional maximal subgroups described
+in Lemma 2.3(4). Then G = GSp4(Fℓ) if and only if there exists matrices X,Y ∈ G such that
+(a) the characteristic polynomial of X is irreducible; and
+(b) traceY ≠ 0 and the characteristic polynomial of Y has a linear factor with multiplicity one.
+Proof. The ‘only if’ direction follows from Proposition 5.1 below, where we show that a nonzero
+proportion of elements of GSp4(Fℓ) satisfy the conditions in (a) and (b).
+Now assume that the group G has elements X and Y as in the statement of the proposition. We
+have to show that G = GSp4(Fℓ). By assumption, G is not a subgroup of a maximal subgroup of
+type (4). For each of the remaining types of maximal subgroups in Lemma 2.3, we will use one of
+the elements X or Y to rule out G being contained in a subgroup of that type.
+(a) By Lemma 2.2 (iv), every element of a subgroup of type (1) has a reducible characteristic
+polynomial. The same is true for elements of type (3) by Lemma 2.4 (2). This is violated by
+the element X, so G cannot be contained in a subgroup of type (1) or type (3).
+(b) Recall the notation used in the description of a type (2) maximal subgroups in Lemma 2.3.
+By Lemma 2.4 1a, every element in Gℓ ∖ Mℓ has trace 0. By Lemma 2.2 (iii), an element with
+irreducible characteristic polynomial automatically has nonzero trace. Hence both X and Y
+have nonzero trace, and so cannot be contained in Gℓ ∖ Mℓ. We now consider two cases
+(i) If the two lines are individually defined over Fℓ, then every element in Mℓ preserves a
+two-dimensional subspace and hence has a reducible characteristic polynomial. This is
+violated by the element X.
+(ii) If the two lines are permuted by GFℓ, then the action of Mℓ on the corresponding subspaces
+V and V ′ are conjugate. Therefore, every Fℓ-rational eigenvalue for the action of Frobp
+on V , also appears as an eigenvalue for the action on V ′, with the same multiplicity. This
+is violated by the element Y .
+Hence G cannot be contained in a maximal subgroup of type (2).
+Since any subgroup of GSp4(Fℓ) that is not contained in a proper maximal subgroup of GSp4(Fℓ)
+must equal GSp4(Fℓ), we are done.
+□
+Remark 13. [AdRK13, Corollary 2.2] gives a very similar criterion for a subgroup G of GSp4(Fℓ)
+to contain Sp4(Fℓ), namely that it contains a transvection, and also an element with irreducible
+characteristic polynomial (and hence automatically nonzero trace).
+4.2. Surjectivity tests.
+4.2.1. Surjectivity test for ℓ = 2.
+Proposition 4.3. Let A be the Jacobian of the hyperelliptic curve y2 + h(x)y = f(x) defined over
+Q. Then ρA,2 is surjective if and only if the size of the Galois group of the splitting field of 4f + h2
+is 720.
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+21
+Proof. This follows from the fact that GSp4(F2) ≅ S6 which is a group of size 720, and that the
+representation ρA,2 is the permutation action of the Galois group on the six roots of 4f + h2.
+□
+4.2.2. Surjectivity tests for ℓ ≠ 2.
+The tests to rule out the exceptional maximal subgroups rely on the existence of the finite lists
+C1920 and C720 (independent of ℓ), and C7,5040 given in Lemma 2.4(3).
+Test 4.4 (Tests for ruling out exceptional maximal subgroups of GSp4(Fℓ) for ℓ ≠ 2).
+Given a polynomial Pp(t) = t4 − apt + bpt2 − papt + p2 and ℓ ≥ 2,
+(i) Pp(t) passes Test 4.4 (i) if ℓ ≡ ±1 (mod 8) or (a2
+p/p,bp/p) mod ℓ lies outside of C1920 mod ℓ.
+(ii) Pp(t) passes Test 4.4 (ii) if ℓ ≡ ±1 (mod 12) or (a2
+p/p,bp/p) mod ℓ lies outside of C720 mod ℓ.
+(iii) Pp(t) passes Test 4.4 (iii) if ℓ ≠ 7 or (a2
+p/p,bp/p) mod ℓ lies outside of C7,5040.
+Test 4.5 (Tests for ruling out non-exceptional maximal subgroups for ℓ ≠ 2).
+Given a polynomial Pp(t) = t4 − apt + bpt2 − papt + p2 and ℓ ≥ 2,
+(i) Pp(t) passes Test 4.5 (i) if Pp(t) modulo ℓ is irreducible.
+(ii) Pp(t) passes Test 4.5 (ii) if Pp(t) modulo ℓ has a linear factor of multiplicity 1 and has nonzero
+trace.
+For any one of the five tests above, say that the test succeeds if a given polynomial Pp(t) passes
+the corresponding test.
+Remark 14. We call an auxiliary prime p a witness for a given prime ℓ if the polynomial Pp(t)
+passes one of our tests for ℓ. The verbose output of our code prints witnesses for each of our tests
+for each prime ℓ in PossiblyNonsurjectivePrimes but not in LikelyNonsurjectivePrimes(B).
+4.3. Justification for surjectivity tests. Considering Tests 4.4 and 4.5, we define
+Cα = {M ∈ GSp4(Fℓ) ∶ PM(t) is irreducible}
+Cβ = {M ∈ GSp4(Fℓ) ∶ tr(M) ≠ 0 and PM(t) has a linear factor of multiplicity 1}
+Cγ1 = {M ∈ GSp4(Fℓ) ∶ ( tr(M)2
+mult(M), mid(M)
+mult(M)) /∈ Cℓ,1920 or ℓ ≡ ±1
+(mod 8)}
+Cγ2 = {M ∈ GSp4(Fℓ) ∶ ( tr(M)2
+mult(M), mid(M)
+mult(M)) /∈ Cℓ,720 or ℓ ≡ ±1
+(mod 12)}
+Cγ3 = {M ∈ GSp4(Fℓ) ∶ ( tr(M)2
+mult(M), mid(M)
+mult(M)) /∈ Cℓ,5040 or ℓ ≠ 7}
+Cγ = Cγ1 ∩ Cγ2 ∩ Cγ3.
+Proof of Theorem 1.1(2) and Theorem 1.3. Let B > 0. Since LikelyNonsurjectivePrimes(B) is a sub-
+list of PossiblyNonsurjectivePrimes, which contains all nonsurjective primes by Theorem 1.1(1), any
+prime not in PossiblyNonsurjectivePrimes is surjective. Now consider ℓ ∈ PossiblyNonsurjectivePrimes
+and not in LikelyNonsurjectivePrimes(B). If ℓ = 2, then by Proposition 4.3, ρA,2 is surjective. If
+ℓ > 2, this means that we found primes p1,p2,p3,p4,p5 ≤ B each distinct from ℓ and of good reduc-
+tion for A for which ρA,ℓ(Frobp1) ∈ Cα, ρA,ℓ(Frobp2) ∈ Cβ, ρA,ℓ(Frobp3) ∈ Cγ1, ρA,ℓ(Frobp4) ∈ Cγ2,
+and ρA,ℓ(Frobp4) ∈ Cγ3. Note that by (1), the similitude factor mult(ρA,ℓ(Frobp)) is p. Therefore,
+by Lemma 2.4(3), it follows that ρA,ℓ(GQ) is not contained in an exceptional maximal subgroup.
+The surjectivity of ρA,ℓ now follows from Proposition 4.2.
+Finally, we will show that if B is sufficiently large (as quantified by Theorem 1.3), then any
+prime ℓ in PossiblyNonsurjectivePrimes is nonsurjective. Since the sets Cα, Cβ, Cγ1, Cγ2 and Cγ3
+are nonempty by Proposition 5.1 below and closed under conjugation, it follows by Lemma 2.10,
+there exist primes p1,p2,p3,p4,p5 ≤ B as above.
+□
+
+22
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+Remark 15. If we assume both GRH and AHC, Ram Murty and Kumar Murty [MM97, p. 52] noted
+(see also [FJ20, Theorem 2.3]) that the bound (4) can be replaced with p ≪ (log dK)2
+∣S∣
+. Proposition
+5.1, which follows, shows that the sets Cα, Cβ, and Cγ have size at least ∣ GSp4(Fℓ)∣
+10
+. This can be
+used to prove the ineffective version of Theorem 1.3 which relies on AHC noted in the introduction
+in a manner similar to the proof of Theorem 1.3.
+5. The probability of success
+In this section we prove Theorem 1.4, by studying the probability that a matrix chosen uniformly
+at random from GSp4(Fℓ) is contained in each of Cα, Cβ, and Cγ defined in Section 4.3. Let αℓ, βℓ,
+and γℓ respectively be the probabilities that a matrix chosen uniformly at random from GSp4(Fℓ)
+is contained in Cα, Cβ, or Cγ.
+Proposition 5.1. Let M be a matrix chosen uniformly at random from GSp4(Fℓ) with ℓ odd. Then
+(i) The probability that M ∈ Cα is given by
+αℓ = 1
+4 −
+1
+2(ℓ2 + 1).
+(ii) The probability that M ∈ Cβ is given by
+βℓ = 3
+8 −
+3
+4(ℓ − 1) +
+1
+2(ℓ − 1)2 .
+(iii) The probability that M ∈ Cγ is
+γℓ ≥ 1 −
+3ℓ
+ℓ2 + 1.
+Remark 16. Magma code that directly verifies the sizes of Cα,Cβ,Cγ (i.e. computes αℓ,βℓ,γℓ) for
+small ℓ may be found in helper_scripts/SanityCheckProbability.m in the repository.
+[Shi82] characterizes all conjugacy classes of elements of GSp4(Fℓ) for ℓ odd, grouping them into
+26 different types. For each type γ, Shinoda further computes the number N(γ) of conjugacy
+classes of type γ and the size of the centralizer ∣CGSp4(Fℓ)(γ)∣, which is the size of the centralizer
+∣CGSp4(Fℓ)(M)∣ of M in GSp4(Fℓ) for any M in a conjugacy class of type γ. The size ∣C(γ)∣ of any
+conjugacy class of type γ can then easily be computed as
+∣C(γ)∣ =
+∣GSp4(Fℓ)∣
+∣CGSp4(Fℓ)(γ)∣
+and the probability that a uniformly chosen M ∈ GSp4(Fℓ) has conjugacy type γ is then given by
+(9)
+N(γ)∣C(γ)∣
+∣GSp4(Fℓ)∣ =
+N(γ)
+∣CGSp4(Fℓ)(γ)∣.
+To prove Proposition 5.1, we will need to examine a handful of types of conjugacy classes of
+GSp4(Fℓ).
+There is only a single conjugacy type γ whose characteristic polynomials are irreducible. This
+type is denoted K0 in [Shi82] where it is shown there that N(K0) = (ℓ−1)(ℓ2−1)
+4
+and ∣CGSp4(Fℓ)(K0)∣ =
+(ℓ − 1)(ℓ2 + 1).
+While there is only one way for a polynomial to be irreducible, there are several ways for a
+quartic polynomial to have a root of odd order. However, only some of these can occur if f(t) is
+the characteristic polynomial of a matrix M ∈ GSp4(Fℓ) and we only need to concern ourselves
+with the following three possibilities:
+(a) f(t) splits completely over Fℓ;
+(b) f(t) has two roots over Fℓ, both of which occur with multiplicity one; and
+(c) f(t) has two simple roots and one double root over Fℓ.
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+23
+Cases (a) and (b) correspond to the conjugacy types H0 and J0 in [Shi82] respectively.
+In
+contrast, there are two types of conjugacy classes for which f(t) has two simple roots and one
+double root, which are denoted E0 and E1 in [Shi82].
+The number of conjugacy classes and centralizer size for each of these conjugacy types is given by
+Table 2, along with the associated probability that a uniform random M ∈ GSp4(Fℓ) has conjugacy
+type γ computed using (9).
+Type γ in [Shi82]
+N(γ)
+∣CGSp4(Fℓ)(γ)∣
+Associated Probability
+K0 (Irreducible)
+(ℓ−1)(ℓ2−1)
+4
+(ℓ2 + 1)(ℓ − 1)
+1
+4 −
+1
+2(ℓ2+1)
+H0 (Split)
+(ℓ−1)(ℓ−3)2
+8
+(ℓ − 1)3
+1
+8 −
+1
+2(ℓ−1) +
+1
+2(ℓ−1)2
+J0 (Two Simple Roots)
+(ℓ−1)3
+4
+(ℓ + 1)(ℓ − 1)2
+1
+4 −
+1
+2(ℓ+1)
+E0 (One Double Root)
+(ℓ−1)(ℓ−3)
+2
+ℓ(ℓ − 1)2(ℓ2 − 1)
+1
+2ℓ(ℓ2−1) −
+1
+ℓ(ℓ−1)(ℓ2−1)
+E1 (One Double Root)
+(ℓ−1)(ℓ−3)
+2
+ℓ(ℓ − 1)2
+1
+2ℓ −
+1
+ℓ(ℓ−1)
+Table 2. Number of conjugacy classes and centralizer sizes for each conjugacy class
+type in [Shi82].
+Proof of Proposition 5.1. Part (i) is simply the entry in Table 2 in the last column corresponding
+to the “K0 (Irreducible)” type.
+We now establish part (ii). As indicated in the discussion above Table 2, the only conjugacy
+classes of matrices in GSp4(Fℓ) whose characteristic polynomials have some linear factors of odd
+multiplicity are those of the types H0,J0,E0,E1. However, for part (ii) since we are only interested
+in matrices M also having non-zero trace, it is insufficient to simply sum over the rightmost entries
+in the bottom four rows of Table 2. From [Shi82, Table 2], we see that the elements of E0 and E1
+have trace c(a+1)2
+a
+for some c,a ∈ F×
+ℓ with a ≠ ±1. In particular, it follows that elements of types E0
+and E1 have nonzero traces. The elements of J0 have trace (c+a)(c+aℓ)
+c
+where c ∈ F×
+ℓ and a ∈ Fℓ2 ∖Fℓ.
+Therefore, the elements of J0 also have nonzero trace.
+It remains to analyze which conjugacy classes of Type H0 have nonzero trace. Following [Shi82],
+the
+(ℓ−1)(ℓ−3)2
+8
+conjugacy classes of type H0 correspond to quadruples of distinct elements in
+a1,a2,b1,b2 ∈ F×
+ℓ satisfying a1b1 = a2b2 modulo the action of swapping any of a1 with b1, a2 with
+b2, or a1,b1 with a2,b2. The eigenvalues of any matrix in the conjugacy class are a1, a2, b1, and b2.
+Consequently the matrix has trace zero only if either a2 = −a1 and b2 = −b1 or b1 = −a2 and b2 = −a1.
+This accounts for (ℓ−1)(ℓ−3)
+4
+of the (ℓ−1)(ℓ−3)2
+8
+conjugacy classes of type H0, leaving (ℓ−1)(ℓ−3)(ℓ−5)
+8
+conjugacy classes with non-zero trace. As a result, the probability that a matrix M ∈ GSp4(Fℓ)
+chosen uniformly at random has non-zero trace and totally split characteristic polynomial is
+(10)
+(ℓ − 1)(ℓ − 3)(ℓ − 5)
+8(ℓ − 1)3
+= 1
+8 −
+3
+4(ℓ − 1) +
+1
+(ℓ − 1)2 .
+To obtain part (ii), we add (10) to the entries in the rightmost column of the final three rows of
+Table 2, getting
+(1
+8 −
+3
+4(ℓ − 1) +
+1
+(ℓ − 1)2 ) + (1
+4 −
+1
+2(ℓ + 1)) + (
+1
+2ℓ(ℓ2 − 1) −
+1
+ℓ(ℓ − 1)(ℓ2 − 1)) + ( 1
+2ℓ −
+1
+ℓ(ℓ − 1))
+= 3
+8 −
+3
+4(ℓ − 1) +
+1
+2(ℓ − 1)2 .
+
+24
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+To prove (iii), we start by noting that for any pair (u,v), the cardinality of the set
+{t4 − at3 + bt2 − amt + m2 ∶ a,b ∈ Fℓ,m ∈ F×
+ℓ and (a2
+m , b
+m) = (u,v)}
+is at most ℓ − 1.
+By [Cha97, Theorem 3.5], the number of matrices in GSp4(Fℓ) with a given
+characteristic polynomial is at most (ℓ+3)8. Assuming ℓ ≠ 7, by combining these observations, and
+noting that ∣Cℓ,720 ∪ Cℓ,1920∣ ≤ 14, we obtain the bound
+γℓ ≥ 1 − 14(ℓ − 1)(ℓ + 3)8
+∣GSp4(Fℓ)∣
+.
+For ℓ > 17, this implies the claimed bound. For 3 ≤ ℓ ≤ 17, we directly check the claim using
+Magma.
+□
+Lemma 5.2. Let C/Q be a typical genus 2 curve with Jacobian A and suppose ℓ is an odd prime
+such that ρA,ℓ is surjective. For any ϵ > 0, there exists an effective constant B0 (with B0 > ℓNA)
+such that for any B > B0 and each δ ∈ {α,β,γ}, we have
+∣∣{p prime ∶ B ≤ p ≤ 2B and ρA,ℓ(Frobp) ∈ Cδ}∣
+∣{p prime ∶ B ≤ p ≤ 2B}∣
+− δℓ∣ < ϵ.
+Proof. Let G = Gal(Q(A[ℓ])/Q) and S ⊆ G be any subset that is closed under conjugation. By
+taking B to be sufficiently large, we have that B > ℓNA and can make
+∣∣{p prime ∶ B ≤ p ≤ 2B and Frobp ∈ S}∣
+∣{p prime ∶ B ≤ p ≤ 2B}∣
+− ∣S∣
+∣G∣∣
+arbitrarily small by (3).
+Moreover, the previous statement can be made effective by using an
+effective version of the Chebotarev density theorem. The result then follows because each of the
+sets Cα, Cβ, and Cγ is closed under conjugation.
+□
+For positive integers n and B > ℓNA, let P(B,n) be the probability that n primes p1,...,pn
+(possibly non-distinct) chosen uniformly at random in the interval [B,2B] have the property that
+ρA,ℓ(Frobpi) /∈ Cα for each i
+or
+ρA,ℓ(Frobpi) /∈ Cβ for each i
+or
+ρA,ℓ(Frobpi) /∈ Cγ for each i.
+Corollary 5.3. Suppose C and ℓ are as in Lemma 5.2 and let n be a positive integer. For any
+ϵ > 0, there exists an effective constant B0 (with B0 > ℓNA) such that for all B > B0, we have
+P(B,n) < (1 − αℓ)n + (1 − βℓ)n + (1 − γℓ)n + ϵ.
+Proof. For δ ∈ {α,β,γ}, let Xδ be the event that none of the ρA,ℓ(Frobpi) are contained in Cδ. We
+then have
+P(Xα ∪ Xβ ∪ Xγ) ≤ P(Xα) + P(Xβ) + P(Xγ)
+The result then follows by Lemma 5.2, which shows that there exists a B0 such that the probabilities
+of Xα, Xβ, and Xγ can be made arbitrarily close to (1−αℓ)n, (1−βℓ)n, and (1−γℓ)n respectively.
+□
+Proof of Theorem 1.4. The claim made by Theorem 1.4 is that P(B,n) < 3⋅( 9
+10)
+n for B sufficiently
+large. By Proposition 5.1, we have 1 − αℓ ≤ 4
+5, 1 − βℓ ≤ 7
+8, and 1 − γℓ ≤ 9
+10 for all ℓ odd. The result
+then follows from Corollary 5.3 because (4
+5)
+n + (7
+8)
+n + ( 9
+10)
+n < 3 ⋅ ( 9
+10)
+n.
+□
+6. Results of computation and interesting examples
+We report on the results of running our algorithm on a dataset of 1,823,592 typical genus 2
+curves with conductor bounded by 220 that are part of a new dataset of approximately 5 million
+curves currently being prepared for addition into the LMFDB. Running our algorithm on all of
+these curves in parallel took about 45 hours on MIT’s Lovelace computer (see the Introduction for
+the hardware specification of this machine).
+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+25
+We first show in Table 3 how many of these curves were nonsurjective at particular primes,
+indicating also if this can be explained by the existence of a rational torsion point of that prime
+order. We found 31 as the largest nonsurjective prime, which occurred for the curve
+(11)
+y2 + (x + 1)y = x5 + 23x4 − 48x3 + 85x2 − 69x + 45
+of conductor 72 ⋅ 312 and discriminant 72 ⋅ 319 (the prime 2 was also nonsurjective here).
+The
+Jacobian of this curve does not admit a nontrivial rational 31-torsion point, so unlike many other
+instances of nonsurjective primes we observed, this one cannot be explained by the presence of
+rational torsion. One could ask if it might be explained by the existence of a Q-rational 31-isogeny
+(as suggested by Algorithm 3.1, since 31 is returned by Algorithm 3.6). This seems to be the case
+- see forthcoming work of van Bommel, Chidambaram, Costa, and Kieffer [vBCCK22] where the
+isogeny class of this curve (among others) is computed.
+nonsurj. prime
+No. of curves w/ torsion
+No. of curves w/o torsion
+Example curve
+2
+1,100,706
+462,616
+464.a.464.1
+3
+79,759
+98,750
+277.a.277.2
+5
+12,040
+10,809
+16108.b.64432.1
+7
+1,966
+2,213
+295.a.295.2
+11
+167
+210
+4288.b.548864.1
+13
+108
+310
+439587.d.439587.1
+17
+22
+61
+1996.b.510976.1
+19
+10
+20
+1468.6012928
+23
+2
+8
+1696.1736704
+29
+1
+5
+976.999424
+31
+0
+1
+47089.1295541485872879
+Table 3. Nonsurjective primes in the dataset, and whether they are explained by
+torsion, with examples from the LMFDB dataset if available, else a string of the
+form “conductor.discrimnant”.
+We also observed (see Table 4) that the vast majority of curves had less than 3 nonsurjective
+primes.
+No. of nonsurj. primes
+No. of curves
+Example curve
+Nonsurj. primes of example
+0
+211,620
+743.a.743.1
+–
+1
+1,455,473
+1923.a.1923.1
+5 (torsion)
+2
+155,186
+976.a.999424.1
+2, 29(torsion)
+3
+1,313
+15876.a.15876.1
+2, 3, 5
+Table 4. Frequency count of nonsurjective primes in the dataset, with examples
+from the LMFDB dataset.
+Instructions for obtaining the entire results file may be found in the README.md file of the
+repository.
+Remark 17. It would be interesting to know if there is a uniform upper bound on the largest prime
+ℓ that could occur as a nonsurjective prime for the Jacobian of a genus 2 curve defined over Q,
+
+26
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+analogous to the conjectural bound of 37 for the largest nonsurjective prime for elliptic curves
+defined over Q (see e.g. [BPR13, Introduction]). As the example of (11) shows, this bound - if it
+exists - would have to be at least 31.
+We conclude with a few examples that illustrate where Algorithm 3.1 fails when the abelian
+surface has extra (geometric) endomorphisms.
+Example 6.1. The Jacobian A of the genus 2 curve 3125.a.3125.1 on the LMFDB given by y2+y =
+x5 has End(AQ) = Z but End(AQ) = Z[ζ5]. Let φ be the Dirichlet character of modulus 5 defined
+by the Legendre symbol
+φ∶(Z/5Z)× → {±1},
+2 ↦ −1.
+In this case, Algorithm 3.13 fails to find an auxilliary prime p < 1000 for which ap ≠ 0 and φ(p) = −1.
+This is consistent with the endomorphism calculation, since the trace of ρA,ℓ(Frobp) is 0 for all
+primes p that do not split completely in Q(ζp) and any inert prime in Q(
+√
+5) automatically does
+not split completely in Q(ζ5).
+Example 6.2. The modular curve X1(13) (169.a.169.1) has genus 2 and its Jacobian J1(13) has
+CM by Z[ζ3] over Q. As in [MT74, Claim 2, page 45], for any prime ℓ that splits as ππ in Q(ζ3), the
+representation J1(13)[ℓ] splits as a direct sum Vπ ⊕Vπ of two 2-dimensional subrepresentations that
+are dual to each other. (A similar statement holds for J1(13)[ℓ]⊗Fℓ Fℓ, and so this representation is
+never absolutely irreducible.) As expected, Algorithm 3.6 fails to find an auxiliary prime p < 1000
+for which Rp is nonzero.
+Example 6.3. The first (ordered by conductor) curve whose Jacobian J admits real multiplication
+over Q is the curve 529.a.529.1; indeed, this Jacobian is isogenous to the Jacobian of the modular
+curve X0(23). Since there is a single Galois orbit of newforms - call it f - of level Γ0(23) and weight
+2, we have that J is isogenous to the abelian variety Af associated to f, and thus we expect the
+integer Mself-dual output by Algorithm 3.10 to be zero for any auxiliary prime, which is indeed the
+case.
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+
+COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
+29
+Appendix A. Exceptional maximal subgroups of GSp4(Fℓ)
+ℓ
+type
+choices
+generators
+ℓ ≡ 5 (mod 8)
+G1920
+b2 = −1 in Fℓ
+⎛
+⎜⎜⎜
+⎝
+1
+0
+0
+−1
+0
+1
+−1
+0
+0
+1
+1
+0
+1
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+1
+0
+0
+b
+0
+1
+b
+0
+0
+b
+1
+0
+b
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+1
+0
+0
+−1
+0
+1
+1
+0
+0
+−1
+1
+0
+1
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+1
+0
+1
+0
+0
+1
+0
+1
+−1
+0
+1
+0
+0
+−1
+0
+1
+⎞
+⎟⎟⎟
+⎠
+ℓ ≡ 3 (mod 8)
+G1920
+b2 = −2 in Fℓ
+⎛
+⎜⎜⎜
+⎝
+1
+0
+0
+−1
+0
+1
+−1
+0
+0
+1
+1
+0
+1
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+0
+0
+0
+b
+0
+0
+b
+0
+0
+b
+2
+0
+b
+0
+0
+2
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+1
+0
+0
+−1
+0
+1
+1
+0
+0
+−1
+1
+0
+1
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+1
+0
+1
+0
+0
+1
+0
+1
+−1
+0
+1
+0
+0
+−1
+0
+1
+⎞
+⎟⎟⎟
+⎠
+ℓ ≡ 7 (mod 12)
+G720
+a2 + a + 1 = 0 in Fℓ
+⎛
+⎜⎜⎜
+⎝
+a
+0
+0
+0
+0
+a
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+a
+0
+0
+0
+0
+1
+0
+0
+0
+0
+a
+0
+0
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+a
+0
+−a − 1
+a + 1
+0
+a
+−a − 1
+−a − 1
+−a − 1
+−a − 1
+−1
+0
+a + 1
+−a − 1
+0
+−1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+0
+−1
+0
+0
+1
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+1
+0
+⎞
+⎟⎟⎟
+⎠
+ℓ ≡ 5 (mod 12)
+G720
+b2 = −1 in Fℓ
+⎛
+⎜⎜⎜
+⎝
+−1
+0
+0
+−1
+0
+−1
+−1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+0
+0
+0
+1
+0
+−1
+−1
+0
+0
+1
+0
+0
+−1
+0
+0
+−1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+−b − 1
+b
+2b
+−2b + 1
+b
+b − 1
+2b + 1
+2b
+b
+b − 1
+−b − 2
+−b
+−b − 1
+b
+−b
+b − 2
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+0
+−b
+−2b
+0
+b
+0
+0
+2b
+−2b
+0
+0
+−b
+0
+2b
+b
+0
+⎞
+⎟⎟⎟
+⎠
+ℓ = 7
+G5040
+a = 2 satisfies
+a2 + a + 1 = 0
+⎛
+⎜⎜⎜
+⎝
+2
+0
+0
+0
+0
+2
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+���
+⎜⎜⎜
+⎝
+2
+0
+0
+0
+0
+1
+0
+0
+0
+0
+2
+0
+0
+0
+0
+1
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+6
+0
+5
+2
+0
+6
+5
+5
+5
+5
+4
+0
+2
+5
+0
+4
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+0
+6
+0
+0
+1
+0
+0
+0
+0
+0
+0
+6
+0
+0
+1
+0
+⎞
+⎟⎟⎟
+⎠
+,
+⎛
+⎜⎜⎜
+⎝
+4
+6
+0
+0
+6
+6
+0
+0
+0
+0
+4
+1
+0
+0
+1
+6
+⎞
+⎟⎟⎟
+⎠
+Table 5. Explicit generators for each exceptional maximal subgroup in GSp4(Fℓ)
+(up to conjugacy). The matrices described in Table 5 depend on an auxiliary choice
+of a parameter denoted either a and b in each case. In each row, any one choice of
+the corresponding a and b satisfying the equations described in the table suffices.
+
+30
+BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
+Barinder S. Banwait, Department of Mathematics & Statistics, Boston University, Boston, MA
+Email address: barinder@bu.edu
+URL: https://barinderbanwait.github.io/
+Armand Brumer, Department of Mathematics, Fordham University, New York, NY
+Email address: brumer@fordham.edu
+Hyun Jong Kim, Department of Mathematics, University of Wisconsin-Madison, Madison, WI
+Email address: hyunjong.kim@math.wisc.edu
+URL: https://sites.google.com/wisc.edu/hyunjongkim
+Zev Klagsbrun, Center for Communications Research, San Diego, CA
+Email address: zdklags@ccr-lajolla.org
+Jacob Mayle, Department of Mathematics, Wake Forest University, Winston-Salem, NC
+Email address: maylej@wfu.edu
+Padmavathi Srinivasan, ICERM, Providence, RI
+Email address: padmavathi srinivasan@brown.edu
+URL: https://padmask.github.io/
+Isabel Vogt, Department of Mathematics, Brown University, Providence, RI
+Email address: ivogt.math@gmail.com
+URL: https://www.math.brown.edu/ivogt/
+