{"paper_meta":{"paper_id":"arxiv:0706.1477","title":"0706.1477","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0706.1477v1 [cs.CC] 11 Jun 2007\nVPSPACE and a transfer theorem over the\ncomplex field\nPascal Koiran and Sylvain Perifel\nLIP⋆, ́Ecole Normale Sup ́erieure de Lyon.\n[Pascal.Koiran,Sylvain.Perifel]@ens-lyon.fr\nJune 2007\nAbstract. We extend the transfer theorem of [14] to the complex field. That is,\nwe investigate the links between the class VPSPACE of families of polynomials\nand the Blum-Shub-Smale model of computation over C. Roughly speaking,\na family of polynomials is in VPSPACE if its coefficients can be computed in\npolynomial space. Our main result is that if (uniform, constant-free) VPSPACE\nfamilies can be evaluated efficiently then the class PARC of decision problems\nthat can be solved in parallel polynomial time over the complex field collapses\nto PC. As a result, one must first be able to show that there are VPSPACE\nfamilies which are hard to evaluate in order to separate PC from NPC, or even\nfrom PARC.\nKeywords: computational complexity, algebraic complexity, Blum-Shub-Smale\nmodel, Valiant’s model.\n1\nIntroduction\nIn algebraic complexity theory, two main categories of problems are studied:\nevaluation and decision problems. The evaluation of the permanent of a matrix\nis a typical example of an evaluation problem, and it is well known that the\npermanent family is complete for the class VNP of “easily definable” polynomial\nfamilies [18]. Deciding whether a system of polynomial equations has a solution\nover C is a typical example of a decision problem. This problem is NP-complete\nin the Blum-Shub-Smale model of computation over the complex field [1,2].\nThe main purpose of this paper is to provide a transfer theorem connecting\nthe complexity of evaluation and decision problems. This paper is therefore in\nthe same spirit as [13] and [14] (see also [4]). In the present paper we work with\nthe class of polynomial families VPSPACE introduced in [14]. Roughly speaking,\na family of polynomials (of possibly exponential degree) is in VPSPACE if its\ncoefficients can be evaluated in polynomial space. For instance, it is shown in [14]\nthat resultants of systems of multivariate polynomial equations form a VPSPACE\nfamily. The main result in [14] was that if (uniform, constant-free) VPSPACE\nfamilies can be evaluated efficiently then the class PARR of decision problems\nthat can be solved in parallel polynomial time over the real numbers collapses\nto PR.\n⋆UMR 5668 ENS Lyon, CNRS, UCBL, INRIA. Research report RR2007-27.\n\n2\nPascal Koiran and Sylvain Perifel\nHere we extend this result to the complex field C. At first glance the result\nseems easier because the order ≤over the reals does not have to be taken into\naccount. The result of [14] indeed makes use of a clever combinatorial lemma\nof [10] on the existence of a vector orthogonal to roughly half a collection of\nvectors. More precisely, it relies on the constructive version of this lemma [6].\nOn the complex field, we do not need this construction.\nBut the lack of an order over C makes another part of the proof more difficult.\nIndeed, over R testing whether a point belongs to a real variety is done by testing\nwhether the sum of the squares of the polynomials is zero, a trick that cannot be\nused over the complex field. Hence one of the main technical developments of this\npaper is to explain how to decide with a small number of tests whether a point\nis in the complex variety defined by an exponential number of polynomials. This\nenables us to follow the nonconstructive proof of [12] for our transfer theorem.\nTherefore, the main result of the present paper is that if (uniform, constant-\nfree) VPSPACE families can be evaluated efficiently then the class PARC of deci-\nsion problems that can be solved in parallel polynomial time over the complex\nfield collapses to PC (this is precisely stated in Theorem 2). The class PARC plays\nroughly the same role in the theory of computation over the complex field as\nPSPACE in discrete complexity theory. In particular, it contains NPC [1] (but the\nproof of this inclusion is much more involved than in the discrete case). It follows\nfrom our main result that in order to separate PC from NPC, or even from PARC,\none must first be able to show that there are VPSPACE families which are hard\nto evaluate. This seems to be a very challenging lower bound problem, but it is\nstill presumably easier than showing that the permanent is hard to evaluate.\nOrganization of the paper. We first recall in Section 2 some usual notions\nand notations concerning algebraic complexity (Valiant’s model, the Blum-Shub-\nSmale model) and quantifier elimination. The class VPSPACE is defined in Sec-\ntion 3 and some properties proved in [14] are given. Section 4 explains how to\ndecide with a polynomial number of VPSPACE tests whether a point belongs to\na variety. The main difficulty here is that the variety is given as a union of an\nexponential number of varieties, each defined by an exponential number of poly-\nnomials. Finally, Section 5 is devoted to the proof of the transfer theorem. Sign\nconditions are the main tool in this section. We show that PARC problems are\ndecided in polynomial time if we allow Uniform VPSPACE0 tests. The transfer\ntheorem follows as a corollary.\n2\nNotations and Preliminaries\n2.1\nThe Blum-Shub-Smale Model\nIn contrast with boolean complexity, algebraic complexity deals with other struc-\ntures than {0, 1}. In this paper we will focus on the complex field (C, +, −, ×, =).\nAlthough the original definitions of Blum, Shub and Smale [2,1] are in terms of\nuniform machines, we will follow [17] by using families of algebraic circuits to\nrecognize languages over C, that is, subsets of C∞= S\nn≥0 Cn.\n\nVPSPACE and a transfer theorem over the complex field\n3\nAn algebraic circuit is a directed acyclic graph whose vertices, called gates,\nhave indegree 0, 1 or 2. An input gate is a vertex of indegree 0. An output gate is\na gate of outdegree 0. We assume that there is only one such gate in the circuit.\nGates of indegree 2 are labelled by a symbol from the set {+, −, ×}. Gates of\nindegree 1, called test gates, are labelled “= 0?”. The size of a circuit C, in\nsymbols |C|, is the number of vertices of the graph.\nA circuit with n input gates computes a function from Cn to C. On input\n ̄u ∈Cn the value returned by the circuit is by definition equal to the value of\nits output gate. The value of a gate is defined in the usual way. Namely, the\nvalue of input gate number i is equal to the i-th input ui. The value of other\ngates is then defined recursively: it is the sum of the values of its entries for a\n+-gate, their difference for a −-gate, their product for a ×-gate. The value taken\nby a test gate is 0 if the value of its entry is ̸= 0 and 1 otherwise. Since we are\ninterested in decision problems, we assume that the output is a test gate: the\nvalue returned by the circuit is therefore 0 or 1.\nThe class PC is the set of languages L ⊆C∞such that there exists a tuple\n ̄a ∈Cp and a P-uniform family of polynomial-size circuits (Cn) satisfying the\nfollowing condition: Cn has exactly n + p inputs, and for any ̄x ∈Cn, ̄x ∈L ⇔\nCn( ̄x, ̄a) = 1. The P-uniformity condition means that Cn can be built in time\npolynomial in n by an ordinary (discrete) Turing machine. Note that ̄a plays the\nrole of the machine constants of [1,2].\nAs in [5], we define the class PARC as the set of languages over C recognized\nby a PSPACE-uniform (or equivalently P-uniform) family of algebraic circuits of\npolynomial depth (and possibly exponential size), with constants ̄a as for PC.\nNote at last that we could also define similar classes without constants ̄a. We\nwill use the superscript 0 to denote these constant-free classes, for instance P0\nC\nand PAR0\nC.\nWe end this section with a theorem on the first-order theory of the complex\nnumbers: quantifiers can be eliminated without much increase of the coefficients\nand degree of the polynomials. We give a weak version of the result of [9]:\nin particular, we do not need efficient elimination algorithms. Note that the\nonly allowed constants in our formulae are 0 and 1 (in particular, only integer\ncoefficients can appear). For notational consistency with the remainding of the\npaper, we denote by 2s, 2d and 22M the number of polynomials, their degree\nand the absolute value of their coefficients respectively. This will simplify the\ncalculations and emphasize that s, d and M will be polynomial. Note furthermore\nthat the polynomial p(n, s, d) in the theorem is independent of the formula φ.\nTheorem 1. Let φ be a first-order formula over (C, 0, 1, +, −, ×, =) of the form\n∀ ̄xψ( ̄x), where ̄x is a tuple of n variables and ψ a quantifier-free formula where\n2s polynomials occur. Suppose that their degrees are bounded by 2d and their\ncoefficients by 22M in absolute value.\nThere exists a polynomial p(n, s, d), independent of φ, such that the formula\nφ is equivalent to a quantifier-free formula ψ in which all polynomials have degree\nless than D(n, s, d) = 2p(n,s,d), and their coefficients are integers strictly bounded\nin absolute value by 22MD(n,s,d).\n\n4\nPascal Koiran and Sylvain Perifel\n2.2\nValiant’s Model\nIn Valiant’s model, one computes polynomials instead of recognizing languages.\nWe thus use arithmetic circuits instead of algebraic circuits. A book-length treat-\nment of this topic can be found in [3].\nAn arithmetic circuit is the same as an algebraic circuit but test gates are not\nallowed. That is to say we have indeterminates x1, . . . , xu(n) as input together\nwith arbitrary constants of C; there are +, −and ×-gates, and we therefore\ncompute multivariate polynomials.\nThe polynomial computed by an arithmetic circuit is defined in the usual way\nby the polynomial computed by its output gate. Thus a family (Cn) of arithmetic\ncircuits computes a family (fn) of polynomials, fn ∈C[x1, . . . , xu(n)]. The class\nVPnb defined in [15] is the set of families (fn) of polynomials computed by a\nfamily (Cn) of polynomial-size arithmetic circuits, i.e., Cn computes fn and there\nexists a polynomial p(n) such that |Cn| ≤p(n) for all n. We will assume without\nloss of generality that the number u(n) of variables is bounded by a polynomial\nfunction of n. The subscript nb indicates that there is no bound on the degree\nof the polynomial, in contrast with the original class VP of Valiant where a\npolynomial bound on the degree of the polynomial computed by the circuit is\nrequired. Note that these definitions are nonuniform. The class Uniform VPnb\nis obtained by adding a condition of polynomial-time uniformity on the circuit\nfamily, as in Section 2.1.\nWe can also forbid constants from our arithmetic circuits in unbounded-\ndegree classes, and define constant-free classes. The only constant allowed is\n1 (in order to allow the computation of constant polynomials). As for classes\nof decision problems, we will use the superscript 0 to indicate the absence of\nconstant: for instance, we will write VP0\nnb (for bounded-degree classes, we are to\nbe more careful: the “formal degree” of the circuits comes into play, see [15,16]).\n3\nThe Class VPSPACE\nThe class VPSPACE was introduced in [14]. Some of its properties are given there\nand a natural example of a VPSPACE family coming from algebraic geometry,\nnamely the resultant of a system of polynomial equations, is provided. In this\nsection, after the definition we give some properties without proof and refer\nto [14] for further details.\n3.1\nDefinition\nWe fix an arbitrary field K. The definition of VPSPACE will be stated in terms\nof coefficient function. A monomial xα1\n1 · · · xαn\nn\nis encoded in binary by α =\n(α1, . . . , αn) and will be written ̄xα.\nDefinition 1. Let (fn) be a family of multivariate polynomials with integer co-\nefficients. The coefficient function of (fn) is the function a whose value on input\n\nVPSPACE and a transfer theorem over the complex field\n5\n(n, α, i) is the i-th bit a(n, α, i) of the coefficient of the monomial ̄xα in fn. Fur-\nthermore, a(n, α, 0) is the sign of the coefficient of the monomial ̄xα. Thus fn\ncan be written as\nfn( ̄x) =\nX\nα\n \n(−1)a(n,α,0) X\ni≥1\na(n, α, i)2i−1 ̄xα \n.\nThe coefficient function is a function a : {0, 1}∗→{0, 1} and can therefore be\nviewed as a language. This allows us to speak of the complexity of the coefficient\nfunction. Note that if K is of characteristic p > 0, then the coefficients of our\npolynomials will be integers modulo p (hence with a constant number of bits).\nIn this paper, we will focus only on the field C (which is of characteristic 0).\nDefinition 2. The class Uniform VPSPACE0 is the set of all families (fn) of\nmultivariate polynomials fn ∈K[x1, . . . , xu(n)] satisfying the following require-\nments:\n1. the number u(n) of variables is polynomially bounded;\n2. the polynomials fn have integer coefficients;\n3. the size of the coefficients of fn is bounded by 2p(n) for some polynomial p;\n4. the degree of fn is bounded by 2p(n) for some polynomial p;\n5. the coefficient function of (fn) is in PSPACE.\nWe have chosen to present only Uniform VPSPACE0, a uniform class without\nconstants, because this is the main object of study in this paper. In keeping\nwith the tradition set by Valiant, however, the class VPSPACE is nonuniform\nand allows for arbitrary constants. See [14] for a precise definition.\n3.2\nAn Alternative Characterization and Some Properties\nLet Uniform VPAR0 be the class of families of polynomials computed by a\nPSPACE-uniform family of constant-free arithmetic circuits of polynomial depth\n(and possibly exponential size). This in fact characterizes Uniform VPSPACE0.\nThe proof is given in [14].\nProposition 1. The two classes Uniform VPSPACE0 and Uniform VPAR0 are\nequal.\nWe see here the similarity with PARC, which by definition are those languages\nrecognized by uniform algebraic circuits of polynomial depth. But of course there\nis no test gate in the arithmetic circuits of Uniform VPAR0.\nWe now turn to some properties of VPSPACE. The following two propositions\ncome from [14]. They stress the unlikeliness of the hypothesis that VPSPACE has\npolynomial-size circuits.\nProposition 2. Assuming the generalized Riemann hypothesis (GRH), VPnb =\nVPSPACE if and only if [P/poly = PSPACE/poly and VP = VNP].\n\n6\nPascal Koiran and Sylvain Perifel\nProposition 3. Uniform VPSPACE0 = Uniform VP0\nnb =⇒PSPACE = P-uniform NC.\nRemark 1. To the authors’ knowledge, the separation “PSPACE ̸= P-uniform NC”\nis not known to hold (by contrast, PSPACE can be separated from logspace-\nuniform NC thanks to the space hierarchy theorem).\nLet us now state the main result of this paper.\nTheorem 2 (main theorem). If Uniform VPSPACE0 = Uniform VP0\nnb then\nPAR0\nC = P0\nC.\nNote that the collapse of the constant-free class PAR0\nC to P0\nC implies PARC =\nPC: just replace constants by new variables so as to transform a PARC problem\ninto a PAR0\nC problem, and then replace these variables by their original values\nso as to transform a P0\nC problem into a PC problem.\nThe next section is devoted to the problem of testing whether a point be-\nlongs to a variety. This problem is useful for the proof of the theorem: indeed,\nfollowing [12], several tests of membership to a variety will be made; the point\nhere is to make them constructive and efficient. The main difficulty is that the\nvariety can be defined by an exponential number of polynomials.\n4\nTesting Membership to a Union of Varieties\nIn this section we explain how to perform in Uniform VPSPACE0 membership\ntests of the form “ ̄x ∈V ”, where V ⊆Cn is a variety. We begin in Section 4.1 by\nthe case where V is given by s polynomials. In that case, we determine after some\nprecomputation whether ̄x ∈V in n + 1 tests. We first need two lemmas given\nbelow in order to reduce the number of polynomials and to replace transcendental\nelements by integers.\nThen, in Section 4.2, we deal with the case where V is given as a union of\nan exponential number of such varieties, as in the actual tests of the algorithm\nof Section 5. Determining whether ̄x ∈V still requires n + 1 tests, but the\nprecomputation is slightly heavier.\nLet us first state two useful lemmas. Suppose a variety V is defined by\nf1, . . . , fs, where fi ∈Z[x1, . . . , xn]. We are to determine whether ̄x ∈V with\nonly n + 1 tests, however big s might be. In a nonconstructive manner, this is\npossible and relies on the following classical lemma already used (and proved)\nin [12]: any n + 1 “generic” linear combinations of the fi also define V (the\nresult holds over any infinite field but here we need it only over C). We state\nthis lemma explicitly since we will also need it in our constructive proof.\nLemma 1. Let f1, . . . , fs ∈Z[x1, . . . , xn] be polynomials and V be the variety\nof Cn they define. Then for all coefficients (αi,j)i=1..s,j=1..n+1 ∈Cs(n+1) alge-\nbraically independent over Q, the n + 1 linear combinations gj = Ps\ni=1 αi,jfi\n(for j from 1 to n + 1) also define V .\n\nVPSPACE and a transfer theorem over the complex field\n7\nUnfortunately, in our case we cannot use transcendental numbers and must\nreplace them by integers. The following lemma from [11] asserts that integers\ngrowing sufficiently fast will do. Once again, this is a weaker version adapted to\nour purpose.\nLemma 2. Let φ(α1, . . . , αr) be a quantifier-free first-order formula over the\nstructure (C, 0, 1, +, −, ×, =), containing only polynomials of degree less than\nD and whose coefficients are integers of absolute value strictly bounded by C.\nAssume furthermore that φ( ̄α) holds for all coefficients ̄α = (α1, . . . , αr) ∈Cr\nalgebraically independent over Q.\nThen φ( ̄β) holds for any sequence (β1, . . . , βr) of integers satisfying β1 ≥C\nand βj+1 ≥CDjβD\nj\n(for 1 ≤j ≤r −1).\nThe proof can be found in [11, Lemma 5.4] and relies on the lack of big\ninteger roots of multivariate polynomials.\nLet us sketch a first attempt to prove a constructive version of Lemma 1,\nnamely that n + 1 polynomials with integer coefficients are enough for defin-\ning V (this first try will not work but gives the idea of the proof of the next\nsection). The idea is to use Lemma 2 with the formula φ( ̄α) that tells us that\nthe n + 1 linear combinations of the fi with αi,j as coefficients define the same\nvariety as f1, . . . , fs. At first this formula is not quantifier-free, but over C we\ncan eliminate quantifiers while keeping degree and coefficients reasonably small\nthanks to Theorem 1. Lemma 1 asserts that φ( ̄α) holds as soon as the αi,j are\nalgebraically independent. Then Lemma 2 tells us that φ( ̄β) holds for integers\nβi,j growing fast enough. Thus V is now defined by n + 1 linear combinations of\nthe fi with integer coefficients.\nIn fact, this strategy fails to work for our purpose because the coefficients\ninvolved are growing too fast to be computed in polynomial space. That is why\nwe will proceed by stages in the proofs below: we adopt a divide-and-conquer\napproach and use induction.\n4.1\nTests of Membership\nThe base case of our induction is the following lemma, whose proof is sketched\nin the end of the preceding section. We only consider here a small number of\npolynomials, therefore avoiding the problem of too big coefficients mentioned in\nthe preceding section.\nLemma 3. There exists a polynomial q(n, d) such that, if V ⊆Cn is a variety\ndefined by 2(n + 1) polynomials f1, . . . , f2(n+1) ∈Z[x1, . . . , xn] of degree ≤2d\nand of coefficients bounded by 22M in absolute value, then:\n1. the variety V is defined by n + 1 polynomials g1, . . . , gn+1 ∈Z[x1, . . . , xn] of\ndegree ≤2d and of coefficients bounded by 22M+q(n,d) in absolute value;\n2. furthermore, the coefficients of the gi are bitwise computable from those of\nthe fj in working space Mq(n, d).\n\n8\nPascal Koiran and Sylvain Perifel\nProof. The first-order formula φ( ̄α) (where ̄α ∈C2(n+1)2), expressing that the\nn + 1 linear combinations of the fj’s with coefficients ̄α also define V , can be\nwritten as follows:\nφ( ̄α) ≡∀x ∈Cn\n \n \nn+1\n^\ni=1\n2(n+1)\nX\nj=1\nαi,jfj(x) = 0 ↔\n2(n+1)\n^\nj=1\nfj(x) = 0\n \n ,\nwhere αi,j is a shorthand for α2(i−1)(n+1)+j. The polynomials in this formula are\nof degree ≤1 + 2d and their coefficients are bounded in absolute value by 22M .\nOver C, the quantifier of this formula can be eliminated by Theorem 1: φ( ̄α)\nis equivalent to a quantifier-free formula ψ( ̄α), the polynomials occuring in which\nhave their degree less than D = D(n, log(3(n + 1)), d + 1) and their coefficients\nstrictly bounded in absolute value by C = 22MD, where D(n, log(3(n + 1)), d +\n1) = 2p(n,log(3(n+1)),d+1) is defined in Theorem 1.\nBy Lemma 1, ψ( ̄α) holds for all coefficients ̄α algebraically independent, so\nthat we wish to apply Lemma 2 with integers βi growing sufficiently fast. Let\nr = (1 + 2(n + 1)2)p(n, log(3(n + 1)), d + 1), so that\nD ≤2r and CD2(n+1)2 ≤22M+r\nand define\nβi = 22M+2ir for 1 ≤i ≤2(n + 1)2.\nNote that for all i, βi ≤22M+4(n+1)2r, and it is furthermore easy to check that\nβ1 ≥C and βi+1 ≥CDiβD\ni . Thus by Lemma 2, ψ( ̄β) is true. Define the poly-\nnomial q(n, d) = 1 + 4(n + 1)2r (up to a multiplicative constant for the space\ncomplexity below). Now, letting\ngi =\n2(n+1)\nX\nj=1\nβi,jfj,\nwhere βi,j is a shorthand for β2(i−1)(n+1)+j, proves the first point of the lemma.\nFor the second point, remark that the coefficients βi are bitwise computable\nin space O(M + rn2) and that the coefficients of the gi are merely a sum of\n2(n + 1) products of βj and coefficients of the fk. This multiplication uses only\nspace O(M + rn2) since the integers involved have encoding size 2O(M+rn2) (in\nour case this is particularly easy because the βj are powers of 2). The 2n + 1\nadditions are also performed in space O(M + rn2). This proves the second point\nof the lemma.\n⊓⊔\nProposition 4 now follows by induction.\nProposition 4. There exists a polynomial p(n, s, d) such that, if V is a vari-\nety defined by 2s polynomials f1, . . . , f2s ∈Z[x1, . . . , xn] of degree ≤2d and of\ncoefficients bounded by 22M in absolute value, then:\n\nVPSPACE and a transfer theorem over the complex field\n9\n1. the variety V is defined by n + 1 polynomials g1, . . . , gn+1 ∈Z[x1, . . . , xn] of\ndegree ≤2d and of coefficients bounded by 22M+p(n,s,d) in absolute value;\n2. moreover, the coefficients of the gi are bitwise computable from those of the\nfj in working space Mp(n, s, d).\nProof. This is done by induction on s. Take p(n, s, d) = sq(n, d) where q(n, d) is\nthe polynomial defined in Lemma 3. The base case 2s ≤2(n + 1) follows from\nLemma 3. Suppose therefore that 2s > 2(n + 1). Call V1 and V2 the varieties\ndefined respectively by f1, . . . , f2s−1 and by f2s−1+1, . . . , f2s. Then V = V1 ∩V2\nand by induction hypothesis, V1 and V2 are both defined by n+1 polynomials of\ndegree ≤2d whose coefficients are bounded by 22M+(s−1)q(n,d) in absolute value\nand computable in space M(s −1)q(n, d).\nTherefore by Lemma 3, V is defined by n + 1 polynomials of degree ≤2d\nwhose coefficients are bounded by 22M+sq(n,d) in absolute value and computable\nin space Msq(n, d) as claimed in the proposition.\n⊓⊔\n4.2\nUnion of Varieties\nIn our case, however, the tests made by the algorithm of Section 5 are not exactly\nof the form studied in the previous section: instead of a single variety given by\ns polynomials, we have to decide “x ∈W?” when W ⊆Cn is the union of k\nvarieties. Of course, since the union is finite W is also a variety, but the encoding\nis not the same as above: now, k sets of s polynomials are given.\nA first naive approach is to define W = ∪iVi by the different products of the\npolynomials defining the Vi, but it turns out that there are too many products to\nbe dealt with. Instead, we will adopt a divide-and-conquer scheme as previously.\nLemma 4. There exists a polynomial q(n, d) such that, if V1 and V2 are two\nvarieties of Cn, each defined by n + 1 polynomials in Z[x1, . . . , xn], respectively\nf1, . . . , fn+1 and g1, . . . , gn+1, of degree ≤2d and of coefficients bounded by 22M\nin absolute value, then:\n1. the variety V = V1 ∪V2 is defined by n + 1 polynomials h1, . . . , hn+1 in\nZ[x1, . . . , xn] of degree ≤2d+1 and of coefficients bounded by 22M+q(n,d) in\nabsolute value;\n2. the coefficients of the hi are bitwise computable from those of the fj and gk\nin space Mq(n, d).\nProof. The variety V is defined by the (n + 1)2 polynomials figj for 1 ≤i, j ≤\nn + 1: these polynomials have degree ≤2d+1. Note moreover that there are at\nmost 2n(d+1) monomials of fixed degree δ ≤2d+1, therefore the coefficients of\nthe figj are a sum of at most 2n(d+1) products of integers of encoding size 2M.\nThus they are computable in space O(Mnd) from those of the fi and gj. This\nalso shows that the coefficients of the products figj are bounded in absolute\nvalue by 2n(d+1)22M+1 ≤22M+1+n(d+1). Applying Proposition 4 now enables to\nconclude if we take q(n, d) = 1 + n(d + 1) + p(n, log((n + 1)2), d + 1), where p is\nthe polynomial defined in Proposition 4.\n⊓⊔\n\n10\nPascal Koiran and Sylvain Perifel\nThe next proposition now follows by induction.\nProposition 5. There exists a polynomial r(n, s, k, d) such that, if V1, . . . , V2k ⊆\nCn are 2k varieties, Vi being defined by 2s polynomials f (i)\n1 , . . . , f (i)\n2s ∈Z[x1, . . . , xn]\nof degree ≤2d and of coefficients bounded by 22M in absolute value, then:\n1. the variety V\n= ∪2k\ni=1Vi is defined by n + 1 polynomials g1, . . . , gn+1 in\nZ[x1, . . . , xn] of degree ≤2d+k and whose coefficients are bounded in ab-\nsolute value by 22M+r(n,s,k,d);\n2. moreover, the coefficients of the gi are bitwise computable from those of the\nf (j)\nj′\nin space Mr(n, s, k, d).\nProof. We proceed by induction on k. Define r(n, s, k, d) = (k+1)(p(n, s, d+k)+\nq(n, d+k)), where p and q are defined in Proposition 4 and Lemma 4 respectively.\nThe base case k = 0 is merely an application of Proposition 4. For k > 0, we first\napply Proposition 4 to the Vi, so that each variety Vi is now defined by n + 1\npolynomials of degree ≤2d and whose coefficients are bounded in absolute value\nby 22M+p(n,s,d) and computable in space Mp(n, s, d). Let us group the varieties\nVi by pairs: call Wi = V2i−1 ∪V2i for 1 ≤i ≤2k−1. There are 2k−1 varieties Wi\nand we have V = ∪iWi. By Lemma 4, each variety Wi is defined by n + 1 poly-\nnomials of degree ≤2d+1, of coefficients of bitsize 2M+p(n,s,d)+q(n,d) and bitwise\ncomputable in space M(p(n, s, d) + q(n, d)). By induction hypothesis at rank\nk −1, V is defined by n+1 polynomials of degree ≤2d+1+(k−1), of coefficients of\nbitsize 2M+p(n,s,d)+q(n,d)+k(p(n,⌈log(n+1)⌉,d+k−1)+q(n,d+k−1)) ≤2M+r(n,s,k,d) and\nbitwise computable in space Mr(n, s, k, d). This proves the proposition.\n⊓⊔\nHere is the main consequence on membership tests to a union of varieties.\nCorollary 1. Let p(n) and q(n) be two polynomials. Suppose (fn( ̄x, ̄y, ̄z)) is a\nUniform VPSPACE0 family with | ̄x| = n, | ̄y| = p(n) and | ̄z| = q(n). For an integer\n0 ≤i < 2p(n), call V (n)\ni\n⊆Cn the variety defined by the polynomials fn( ̄x, i, j)\nfor 0 ≤j < 2q(n) (in this notation, i and j are encoded in binary).\nThen there exists a Uniform VPSPACE0 family gn( ̄x, ̄y, ̄z), where | ̄x| = n,\n| ̄y| = p(n) and | ̄z| = ⌈log(n + 1)⌉, such that\n∀ ̄x ∈Cn,\n∀k < 2p(n),\n \n ̄x ∈\nk[\ni=0\nV (n)\ni\n⇐⇒\nn\n^\nj=0\ngn( ̄x, k, j) = 0\n \n .\nProof. If (fn) is a Uniform VPSPACE0 family, by definition there exists a poly-\nnomial p(n) such that the degree of fn is bounded by 2p(n) and the absolute\nvalue of the coefficients by 22p(n). Therefore d, M, s and k are polynomially\nbounded in Proposition 5 and the space needed to compute the coefficients of\ngn is polynomial.\n⊓⊔\n\nVPSPACE and a transfer theorem over the complex field\n11\n5\nProof of the Main Theorem\nSign conditions are the main ingredient of the proof. Over C, we define the\n“sign” of a ∈C by 0 if a = 0 and 1 otherwise. Let us fix a family of polynomials\nf1, . . . , fs ∈Z[x1, . . . , xn]. A sign condition is an element S ∈{0, 1}s. Hence\nthere are 2s sign conditions. Intuitively, the i-th component of a sign condition\ndetermines the sign of the polynomial fi.\n5.1\nSatisfiable Sign Conditions\nThe sign condition of a point ̄x ∈Cn is the tuple S ̄x ∈{0, 1}s defined by\nS ̄x\ni = 0 ⇐⇒fi( ̄x) = 0. We say that a sign condition is satisfiable if it is the\nsign condition of some ̄x ∈Cn. As 0-1 tuples, sign conditions can be viewed as\nsubsets of {1, . . ., s}. Using a fast parallel sorting algorithm (e.g. Cole’s, [7]),\nwe can sort satisfiable sign conditions in polylogarithmic parallel time in a way\ncompatible with set inclusion (e.g. the lexicographic order). We now fix such\na compatible linear order on sign conditions and consider our satisfiable sign\nconditions S(1) < S(2) < . . . < S(N) sorted accordingly.\nThe key point resides in the following theorem, coming from the algorithm\nof [9]: there is a “small” number of satisfiable sign conditions and enumerating\nthem is “easy”.\nTheorem 3. Let f1, . . . , fs ∈Z[x1, . . . , xn] and d be their maximal degree. Then\nthe number of satisfiable sign conditions is N = (sd)O(n), and there is a uniform\nalgorithm working in space\n n log(sd)\n O(1) which, on boolean input f1, . . . , fs (in\ndense representation) and (i, j) in binary, returns the j-th component of the i-th\nsatisfiable sign condition.\nWhen log(sd) is polynomial in n, as will be the case, this yields a PSPACE\nalgorithm. If furthermore the coefficients of fi are computable in polynomial\nspace, we will then be able to use the satisfiable sign conditions in the coefficients\nof VPSPACE families, as in Lemma 5 below.\nLet us explain why we are interested in sign conditions. An arithmetic circuit\nperforms tests of the form f( ̄x) = 0 on input ̄x ∈Cn, where f is a polynomial.\nSuppose f1, . . . , fs is the list of all polynomials that can be tested in any possible\ncomputation. Then two elements of Cn with the same sign condition are simul-\ntaneously accepted or rejected by the circuit: the results of the tests are indeed\nalways the same for both elements.\nThus, instead of finding out whether ̄x ∈Cn is accepted by the circuit, it is\nenough to find out whether the sign condition of ̄x is accepted. The advantage\nresides in handling only boolean tuples (the sign conditions) instead of complex\nnumbers (the input ̄x). But we have to be able to find the sign condition of\nthe input ̄x. This requires first the enumeration of all the polynomials possibly\ntested in any computation of the circuit.\n\n12\nPascal Koiran and Sylvain Perifel\n5.2\nEnumerating all Possibly Tested Polynomials\nIn the execution of an algebraic circuit, the values of some polynomials at the\ninput ̄x are tested to zero. In order to find the sign condition of the input ̄x, we\nhave to be able to enumerate in polynomial space all the polynomials that can\never be tested to zero in the computations of an algebraic circuit. This is done\nlevel by level as in [8, Th. 3] and [14].\nProposition 6. Let C be a constant-free algebraic circuit with n variables and\nof depth d.\n1. The number of different polynomials possibly tested to zero in the computa-\ntions of C is 2d2O(n).\n2. There exists an algorithm using work space (nd)O(1) which, on input C and\nintegers (i, j) in binary, outputs the j-th bit of the representation of the i-th\npolynomial.\nTogether with Theorem 3, this enables us to prove the following result which\nwill be useful in the proof of Proposition 7: in Uniform VPSPACE0 we can enu-\nmerate the polynomials as well as the satisfiable sign conditions.\nLemma 5. Let (Cn) be a uniform family of polynomial-depth algebraic circuits\nwith polynomially many inputs. Call d(n) the depth of Cn and i(n) the number\nof inputs. Let f (n)\n1\n, . . . , f (n)\ns\nbe all the polynomials possibly tested to zero by Cn\nas in Proposition 6, where s = 2O(nd(n)2). There are therefore N = 2O(n2d(n)2)\nsatisfiable sign conditions S(1), . . . , S(N) by Theorem 3.\nThen there exists a Uniform VPSPACE0 family (gn( ̄x, ̄y, ̄z)), where | ̄x| = i(n),\n| ̄y| = O(n2d(n)2) and | ̄z| = O(nd(n)2), such that for all 1 ≤i ≤N and 1 ≤j ≤\ns, we have:\ngn( ̄x, i, j) =\n(\n0\nif S(i)\nj\n= 1\nf (n)\nj\n( ̄x) otherwise.\n5.3\nFinding the Sign Condition of the Input\nIn order to find the sign condition S ̄x of the input ̄x ∈Cn, we will give a\npolynomial-time algorithm which tests some VPSPACE family for zero. Here is\nthe formalized notion of a polynomial-time algorithm with VPSPACE tests.\nDefinition 3. A polynomial-time algorithm with Uniform VPSPACE0 tests is\na Uniform VPSPACE0 family (fn(x1, . . . , xu(n))) together with a uniform fam-\nily (Cn) of constant-free polynomial-size algebraic circuits endowed with spe-\ncial test gates of indegree u(n), whose value is 1 on input (a1, . . . , au(n)) if\nfn(a1, . . . , au(n)) = 0 and 0 otherwise.\nObserve that a constant number of Uniform VPSPACE0 families can be used in\nthe preceding definition instead of only one: it is enough to combine them all in\none by using “selection variables”.\n\nVPSPACE and a transfer theorem over the complex field\n13\nThe precise result we show now is the following. By the “rank” of a satisfiable\nsign condition, we merely mean its index in the fixed order on satisfiable sign\nconditions.\nProposition 7. Let (Cn) be a uniform family of algebraic circuits of polynomial\ndepth and with a polynomial number i(n) of inputs. There exists a polynomial-\ntime algorithm with Uniform VPSPACE0 tests which, on input ̄x ∈Ci(n), returns\nthe rank i of the sign condition S(i) of ̄x with respect to the polynomials g1, . . . , gs\ntested to zero by Cn given by Proposition 6.\nProof. Take the Uniform VPSPACE0 family (gn( ̄x, ̄y, ̄z)) as in Lemma 5: in essence,\ngn enumerates all the polynomials f1, . . . , fs possibly tested to zero in Cn and\nenumerates the N satisfiable sign conditions S(1) < . . . < S(N). The idea now is\nto perform a binary search in order to find the rank i of the sign condition of\nthe input ̄x.\nLet S(j) ∈{0, 1}s be a satisfiable sign condition. We say that S(j) is a candi-\ndate whenever ∀m ≤s, S(j)\nm = 0 ⇒fm( ̄x) = 0. Remark that the sign condition\nof ̄x is the smallest candidate. Call Vj the variety defined by the polynomi-\nals {fm|S(j)\nm\n= 0}: by definition of gn, Vj is also defined by the polynomials\ngn( ̄x, j, k) for k = 1 to s. Note that S(j) is a candidate if and only if ̄x ∈Vj.\nCorollary 1 combined with Lemma 5 asserts that tests of the form ̄x ∈∪k≤jVk\nare in Uniform VPSPACE0. They are used to perform a binary search by making\nj vary. In a number of steps logarithmic in N (i.e. polynomial in n), we find the\nrank i of the sign condition of ̄x.\n⊓⊔\n5.4\nA Polynomial-time Algorithm for PARC Problems\nLemma 6. Let (Cn) be a uniform family of constant-free polynomial-depth al-\ngebraic circuits. There is a (boolean) algorithm using work space polynomial in\nn which, on input i, decides whether the elements of the i-th satisfiable sign\ncondition S(i) are accepted by the circuit Cn.\nProof. We follow the circuit Cn level by level. For test gates, we compute the\npolynomial f to be tested. Then we enumerate the polynomials f1, . . . , fs as\nin Proposition 6 for the circuit Cn and we find the index j of f in this list.\nBy consulting the j-th bit of the i-th satisfiable sign condition with respect to\nf1, . . . , fs (which is done by the polynomial-space algorithm of Theorem 3), we\ntherefore know the result of the test and can go on like this until the output\ngate.\n⊓⊔\nTheorem 4. Let A ∈PAR0\nC. There exists a polynomial-time algorithm with\nUniform VPSPACE0 tests that decides A.\nProof. A is decided by a uniform family (Cn) of constant-free polynomial-depth\nalgebraic circuits. On input ̄x, thanks to Proposition 7 we first find the rank i of\nthe sign condition of ̄x with respect to the polynomials f1, . . . , fs of Proposition 6.\nThen we conclude by a last Uniform VPSPACE0 test simulating the polynomial-\nspace algorithm of Lemma 6 on input i.\n⊓⊔\n\n14\nPascal Koiran and Sylvain Perifel\nTheorem 2 follows immediately from this result. One could obtain other ver-\nsions of these two results by changing the uniformity conditions or the role of\nconstants.\nReferences\n1. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation.\nSpringer-Verlag, 1998.\n2. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over\nthe real numbers: NP-completeness, recursive functions and universal machines.\nBulletin of the American Mathematical Society, 21(1):1–46, 1989.\n3. P. B ̈urgisser. Completeness and Reduction in Algebraic Complexity Theory, vol-\nume 7 of Algorithms and Computation in Mathematics. Springer, 2000.\n4. P. B ̈urgisser. On implications between P-NP-hypotheses: Decision versus compu-\ntation in algebraic complexity. In Mathematical Foundations of Computer Science,\nvolume 2316 of Lecture Notes in Computer Science, pages 3–17. Springer, 2001.\n5. O. Chapuis and P. Koiran. Saturation and stability in the theory of computation\nover the reals. Annals of Pure and Applied Logic, 99:1–49, 1999.\n6. P. Charbit, E. Jeandel, P. Koiran, S. Perifel, and S. Thomass ́e. Finding a vector\northogonal to roughly half a collection of vectors. Available from http://perso.ens-\nlyon.fr/pascal.koiran/publications.html. Accepted for publication in Journal of\nComplexity, 2006.\n7. R. Cole. Parallel merge sort. SIAM J. Comput., 17(4):770–785, 1988.\n8. F. Cucker and D. Grigoriev. On the power of real Turing machines over binary\ninputs. SIAM Journal on Computing, 26(1):243–254, 1997.\n9. N. Fitchas, A. Galligo, and J. Morgenstern. Precise sequential and parallel com-\nplexity bounds for quantifier elimination over algebraically closed fields. Journal\nof Pure and Applied Algebra, 67:1–14, 1990.\n10. D. Grigoriev. Topological complexity of the range searching. Journal of Complexity,\n16:50–53, 2000.\n11. P. Koiran. Randomized and deterministic algorithms for the dimension of algebraic\nvarieties. In Proc. 38th IEEE Symposium on Foundations of Computer Science,\npages 36–45, 1997.\n12. P. Koiran. Circuits versus trees in algebraic complexity. In Proc. STACS 2000,\nvolume 1770 of Lecture Notes in Computer Science, pages 35–52. Springer, 2000.\n13. P. Koiran and S. Perifel.\nValiant’s model: from exponential sums to exponen-\ntial products. In Mathematical Foundations of Computer Science, volume 4162 of\nLecture Notes in Computer Science, pages 596–607. Springer-Verlag, 2006.\n14. P. Koiran and S. Perifel. VPSPACE and a transfer theorem over the reals. In\nProc. 24th Symposium on Theoretical Aspects of Computer Science, volume 4393\nof Lecture Notes in Computer Science, pages 417–428, 2007. Long version available\nfrom http://perso.ens-lyon.fr/pascal.koiran/publications.html.\n15. G. Malod. Polynˆomes et coefficients. PhD thesis, Universit ́e Claude Bernard Lyon\n1, July 2003. Available from http://tel.archives-ouvertes.fr/tel-00087399.\n16. G. Malod and N. Portier. Characterizing Valiant’s algebraic complexity classes.\nIn Mathematical Foundations of Computer Science, volume 4162 of Lecture Notes\nin Computer Science, pages 704–716. Springer-Verlag, 2006.\n17. B. Poizat. Les petits cailloux. Al ́eas, 1995.\n18. L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on\nTheory of Computing, pages 249–261, 1979.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0706.1477v1 [cs.CC] 11 Jun 2007\nVPSPACE and a transfer theorem over the\ncomplex field\nPascal Koiran and Sylvain Perifel\nLIP⋆, ́Ecole Normale Sup ́erieure de Lyon.\n[Pascal.Koiran,Sylvain.Perifel]@ens-lyon.fr\nJune 2007\nAbstract. We extend the transfer theorem of [14] to the complex field. That is,\nwe investigate the links between the class VPSPACE of families of polynomials\nand the Blum-Shub-Smale model of computation over C. Roughly speaking,\na family of polynomials is in VPSPACE if its coefficients can be computed in\npolynomial space. Our main result is that if (uniform, constant-free) VPSPACE\nfamilies can be evaluated efficiently then the class PARC of decision problems\nthat can be solved in parallel polynomial time over the complex field collapses\nto PC. As a result, one must first be able to show that there are VPSPACE\nfamilies which are hard to evaluate in order to separate PC from NPC, or even\nfrom PARC.\nKeywords: computational complexity, algebraic complexity, Blum-Shub-Smale\nmodel, Valiant’s model.\n1\nIntroduction\nIn algebraic complexity theory, two main categories of problems are studied:\nevaluation and decision problems. The evaluation of the permanent of a matrix\nis a typical example of an evaluation problem, and it is well known that the\npermanent family is complete for the class VNP of “easily definable” polynomial\nfamilies [18]. Deciding whether a system of polynomial equations has a solution\nover C is a typical example of a decision problem. This problem is NP-complete\nin the Blum-Shub-Smale model of computation over the complex field [1,2].\nThe main purpose of this paper is to provide a transfer theorem connecting\nthe complexity of evaluation and decision problems. This paper is therefore in\nthe same spirit as [13] and [14] (see also [4]). In the present paper we work with\nthe class of polynomial families VPSPACE introduced in [14]. Roughly speaking,\na family of polynomials (of possibly exponential degree) is in VPSPACE if its\ncoefficients can be evaluated in polynomial space. For instance, it is shown in [14]\nthat resultants of systems of multivariate polynomial equations form a VPSPACE\nfamily. The main result in [14] was that if (uniform, constant-free) VPSPACE\nfamilies can be evaluated efficiently then the class PARR of decision problems\nthat can be solved in parallel polynomial time over the real numbers collapses\nto PR.\n⋆UMR 5668 ENS Lyon, CNRS, UCBL, INRIA. Research report RR2007-27."},{"paragraph_id":"p2","order":2,"text":"2\nPascal Koiran and Sylvain Perifel\nHere we extend this result to the complex field C. At first glance the result\nseems easier because the order ≤over the reals does not have to be taken into\naccount. The result of [14] indeed makes use of a clever combinatorial lemma\nof [10] on the existence of a vector orthogonal to roughly half a collection of\nvectors. More precisely, it relies on the constructive version of this lemma [6].\nOn the complex field, we do not need this construction.\nBut the lack of an order over C makes another part of the proof more difficult.\nIndeed, over R testing whether a point belongs to a real variety is done by testing\nwhether the sum of the squares of the polynomials is zero, a trick that cannot be\nused over the complex field. Hence one of the main technical developments of this\npaper is to explain how to decide with a small number of tests whether a point\nis in the complex variety defined by an exponential number of polynomials. This\nenables us to follow the nonconstructive proof of [12] for our transfer theorem.\nTherefore, the main result of the present paper is that if (uniform, constant-\nfree) VPSPACE families can be evaluated efficiently then the class PARC of deci-\nsion problems that can be solved in parallel polynomial time over the complex\nfield collapses to PC (this is precisely stated in Theorem 2). The class PARC plays\nroughly the same role in the theory of computation over the complex field as\nPSPACE in discrete complexity theory. In particular, it contains NPC [1] (but the\nproof of this inclusion is much more involved than in the discrete case). It follows\nfrom our main result that in order to separate PC from NPC, or even from PARC,\none must first be able to show that there are VPSPACE families which are hard\nto evaluate. This seems to be a very challenging lower bound problem, but it is\nstill presumably easier than showing that the permanent is hard to evaluate.\nOrganization of the paper. We first recall in Section 2 some usual notions\nand notations concerning algebraic complexity (Valiant’s model, the Blum-Shub-\nSmale model) and quantifier elimination. The class VPSPACE is defined in Sec-\ntion 3 and some properties proved in [14] are given. Section 4 explains how to\ndecide with a polynomial number of VPSPACE tests whether a point belongs to\na variety. The main difficulty here is that the variety is given as a union of an\nexponential number of varieties, each defined by an exponential number of poly-\nnomials. Finally, Section 5 is devoted to the proof of the transfer theorem. Sign\nconditions are the main tool in this section. We show that PARC problems are\ndecided in polynomial time if we allow Uniform VPSPACE0 tests. The transfer\ntheorem follows as a corollary.\n2\nNotations and Preliminaries\n2.1\nThe Blum-Shub-Smale Model\nIn contrast with boolean complexity, algebraic complexity deals with other struc-\ntures than {0, 1}. In this paper we will focus on the complex field (C, +, −, ×, =).\nAlthough the original definitions of Blum, Shub and Smale [2,1] are in terms of\nuniform machines, we will follow [17] by using families of algebraic circuits to\nrecognize languages over C, that is, subsets of C∞= S\nn≥0 Cn."},{"paragraph_id":"p3","order":3,"text":"VPSPACE and a transfer theorem over the complex field\n3\nAn algebraic circuit is a directed acyclic graph whose vertices, called gates,\nhave indegree 0, 1 or 2. An input gate is a vertex of indegree 0. An output gate is\na gate of outdegree 0. We assume that there is only one such gate in the circuit.\nGates of indegree 2 are labelled by a symbol from the set {+, −, ×}. Gates of\nindegree 1, called test gates, are labelled “= 0?”. The size of a circuit C, in\nsymbols |C|, is the number of vertices of the graph.\nA circuit with n input gates computes a function from Cn to C. On input\n ̄u ∈Cn the value returned by the circuit is by definition equal to the value of\nits output gate. The value of a gate is defined in the usual way. Namely, the\nvalue of input gate number i is equal to the i-th input ui. The value of other\ngates is then defined recursively: it is the sum of the values of its entries for a\n+-gate, their difference for a −-gate, their product for a ×-gate. The value taken\nby a test gate is 0 if the value of its entry is ̸= 0 and 1 otherwise. Since we are\ninterested in decision problems, we assume that the output is a test gate: the\nvalue returned by the circuit is therefore 0 or 1.\nThe class PC is the set of languages L ⊆C∞such that there exists a tuple\n ̄a ∈Cp and a P-uniform family of polynomial-size circuits (Cn) satisfying the\nfollowing condition: Cn has exactly n + p inputs, and for any ̄x ∈Cn, ̄x ∈L ⇔\nCn( ̄x, ̄a) = 1. The P-uniformity condition means that Cn can be built in time\npolynomial in n by an ordinary (discrete) Turing machine. Note that ̄a plays the\nrole of the machine constants of [1,2].\nAs in [5], we define the class PARC as the set of languages over C recognized\nby a PSPACE-uniform (or equivalently P-uniform) family of algebraic circuits of\npolynomial depth (and possibly exponential size), with constants ̄a as for PC.\nNote at last that we could also define similar classes without constants ̄a. We\nwill use the superscript 0 to denote these constant-free classes, for instance P0\nC\nand PAR0\nC.\nWe end this section with a theorem on the first-order theory of the complex\nnumbers: quantifiers can be eliminated without much increase of the coefficients\nand degree of the polynomials. We give a weak version of the result of [9]:\nin particular, we do not need efficient elimination algorithms. Note that the\nonly allowed constants in our formulae are 0 and 1 (in particular, only integer\ncoefficients can appear). For notational consistency with the remainding of the\npaper, we denote by 2s, 2d and 22M the number of polynomials, their degree\nand the absolute value of their coefficients respectively. This will simplify the\ncalculations and emphasize that s, d and M will be polynomial. Note furthermore\nthat the polynomial p(n, s, d) in the theorem is independent of the formula φ.\nTheorem 1. Let φ be a first-order formula over (C, 0, 1, +, −, ×, =) of the form\n∀ ̄xψ( ̄x), where ̄x is a tuple of n variables and ψ a quantifier-free formula where\n2s polynomials occur. Suppose that their degrees are bounded by 2d and their\ncoefficients by 22M in absolute value.\nThere exists a polynomial p(n, s, d), independent of φ, such that the formula\nφ is equivalent to a quantifier-free formula ψ in which all polynomials have degree\nless than D(n, s, d) = 2p(n,s,d), and their coefficients are integers strictly bounded\nin absolute value by 22MD(n,s,d)."},{"paragraph_id":"p4","order":4,"text":"4\nPascal Koiran and Sylvain Perifel\n2.2\nValiant’s Model\nIn Valiant’s model, one computes polynomials instead of recognizing languages.\nWe thus use arithmetic circuits instead of algebraic circuits. A book-length treat-\nment of this topic can be found in [3].\nAn arithmetic circuit is the same as an algebraic circuit but test gates are not\nallowed. That is to say we have indeterminates x1, . . . , xu(n) as input together\nwith arbitrary constants of C; there are +, −and ×-gates, and we therefore\ncompute multivariate polynomials.\nThe polynomial computed by an arithmetic circuit is defined in the usual way\nby the polynomial computed by its output gate. Thus a family (Cn) of arithmetic\ncircuits computes a family (fn) of polynomials, fn ∈C[x1, . . . , xu(n)]. The class\nVPnb defined in [15] is the set of families (fn) of polynomials computed by a\nfamily (Cn) of polynomial-size arithmetic circuits, i.e., Cn computes fn and there\nexists a polynomial p(n) such that |Cn| ≤p(n) for all n. We will assume without\nloss of generality that the number u(n) of variables is bounded by a polynomial\nfunction of n. The subscript nb indicates that there is no bound on the degree\nof the polynomial, in contrast with the original class VP of Valiant where a\npolynomial bound on the degree of the polynomial computed by the circuit is\nrequired. Note that these definitions are nonuniform. The class Uniform VPnb\nis obtained by adding a condition of polynomial-time uniformity on the circuit\nfamily, as in Section 2.1.\nWe can also forbid constants from our arithmetic circuits in unbounded-\ndegree classes, and define constant-free classes. The only constant allowed is\n1 (in order to allow the computation of constant polynomials). As for classes\nof decision problems, we will use the superscript 0 to indicate the absence of\nconstant: for instance, we will write VP0\nnb (for bounded-degree classes, we are to\nbe more careful: the “formal degree” of the circuits comes into play, see [15,16]).\n3\nThe Class VPSPACE\nThe class VPSPACE was introduced in [14]. Some of its properties are given there\nand a natural example of a VPSPACE family coming from algebraic geometry,\nnamely the resultant of a system of polynomial equations, is provided. In this\nsection, after the definition we give some properties without proof and refer\nto [14] for further details.\n3.1\nDefinition\nWe fix an arbitrary field K. The definition of VPSPACE will be stated in terms\nof coefficient function. A monomial xα1\n1 · · · xαn\nn\nis encoded in binary by α =\n(α1, . . . , αn) and will be written ̄xα.\nDefinition 1. Let (fn) be a family of multivariate polynomials with integer co-\nefficients. The coefficient function of (fn) is the function a whose value on input"},{"paragraph_id":"p5","order":5,"text":"VPSPACE and a transfer theorem over the complex field\n5\n(n, α, i) is the i-th bit a(n, α, i) of the coefficient of the monomial ̄xα in fn. Fur-\nthermore, a(n, α, 0) is the sign of the coefficient of the monomial ̄xα. Thus fn\ncan be written as\nfn( ̄x) =\nX\nα"},{"paragraph_id":"p6","order":6,"text":"(−1)a(n,α,0) X\ni≥1\na(n, α, i)2i−1 ̄xα \n.\nThe coefficient function is a function a : {0, 1}∗→{0, 1} and can therefore be\nviewed as a language. This allows us to speak of the complexity of the coefficient\nfunction. Note that if K is of characteristic p > 0, then the coefficients of our\npolynomials will be integers modulo p (hence with a constant number of bits).\nIn this paper, we will focus only on the field C (which is of characteristic 0).\nDefinition 2. The class Uniform VPSPACE0 is the set of all families (fn) of\nmultivariate polynomials fn ∈K[x1, . . . , xu(n)] satisfying the following require-\nments:\n1. the number u(n) of variables is polynomially bounded;\n2. the polynomials fn have integer coefficients;\n3. the size of the coefficients of fn is bounded by 2p(n) for some polynomial p;\n4. the degree of fn is bounded by 2p(n) for some polynomial p;\n5. the coefficient function of (fn) is in PSPACE.\nWe have chosen to present only Uniform VPSPACE0, a uniform class without\nconstants, because this is the main object of study in this paper. In keeping\nwith the tradition set by Valiant, however, the class VPSPACE is nonuniform\nand allows for arbitrary constants. See [14] for a precise definition.\n3.2\nAn Alternative Characterization and Some Properties\nLet Uniform VPAR0 be the class of families of polynomials computed by a\nPSPACE-uniform family of constant-free arithmetic circuits of polynomial depth\n(and possibly exponential size). This in fact characterizes Uniform VPSPACE0.\nThe proof is given in [14].\nProposition 1. The two classes Uniform VPSPACE0 and Uniform VPAR0 are\nequal.\nWe see here the similarity with PARC, which by definition are those languages\nrecognized by uniform algebraic circuits of polynomial depth. But of course there\nis no test gate in the arithmetic circuits of Uniform VPAR0.\nWe now turn to some properties of VPSPACE. The following two propositions\ncome from [14]. They stress the unlikeliness of the hypothesis that VPSPACE has\npolynomial-size circuits.\nProposition 2. Assuming the generalized Riemann hypothesis (GRH), VPnb =\nVPSPACE if and only if [P/poly = PSPACE/poly and VP = VNP]."},{"paragraph_id":"p7","order":7,"text":"6\nPascal Koiran and Sylvain Perifel\nProposition 3. Uniform VPSPACE0 = Uniform VP0\nnb =⇒PSPACE = P-uniform NC.\nRemark 1. To the authors’ knowledge, the separation “PSPACE ̸= P-uniform NC”\nis not known to hold (by contrast, PSPACE can be separated from logspace-\nuniform NC thanks to the space hierarchy theorem).\nLet us now state the main result of this paper.\nTheorem 2 (main theorem). If Uniform VPSPACE0 = Uniform VP0\nnb then\nPAR0\nC = P0\nC.\nNote that the collapse of the constant-free class PAR0\nC to P0\nC implies PARC =\nPC: just replace constants by new variables so as to transform a PARC problem\ninto a PAR0\nC problem, and then replace these variables by their original values\nso as to transform a P0\nC problem into a PC problem.\nThe next section is devoted to the problem of testing whether a point be-\nlongs to a variety. This problem is useful for the proof of the theorem: indeed,\nfollowing [12], several tests of membership to a variety will be made; the point\nhere is to make them constructive and efficient. The main difficulty is that the\nvariety can be defined by an exponential number of polynomials.\n4\nTesting Membership to a Union of Varieties\nIn this section we explain how to perform in Uniform VPSPACE0 membership\ntests of the form “ ̄x ∈V ”, where V ⊆Cn is a variety. We begin in Section 4.1 by\nthe case where V is given by s polynomials. In that case, we determine after some\nprecomputation whether ̄x ∈V in n + 1 tests. We first need two lemmas given\nbelow in order to reduce the number of polynomials and to replace transcendental\nelements by integers.\nThen, in Section 4.2, we deal with the case where V is given as a union of\nan exponential number of such varieties, as in the actual tests of the algorithm\nof Section 5. Determining whether ̄x ∈V still requires n + 1 tests, but the\nprecomputation is slightly heavier.\nLet us first state two useful lemmas. Suppose a variety V is defined by\nf1, . . . , fs, where fi ∈Z[x1, . . . , xn]. We are to determine whether ̄x ∈V with\nonly n + 1 tests, however big s might be. In a nonconstructive manner, this is\npossible and relies on the following classical lemma already used (and proved)\nin [12]: any n + 1 “generic” linear combinations of the fi also define V (the\nresult holds over any infinite field but here we need it only over C). We state\nthis lemma explicitly since we will also need it in our constructive proof.\nLemma 1. Let f1, . . . , fs ∈Z[x1, . . . , xn] be polynomials and V be the variety\nof Cn they define. Then for all coefficients (αi,j)i=1..s,j=1..n+1 ∈Cs(n+1) alge-\nbraically independent over Q, the n + 1 linear combinations gj = Ps\ni=1 αi,jfi\n(for j from 1 to n + 1) also define V ."},{"paragraph_id":"p8","order":8,"text":"VPSPACE and a transfer theorem over the complex field\n7\nUnfortunately, in our case we cannot use transcendental numbers and must\nreplace them by integers. The following lemma from [11] asserts that integers\ngrowing sufficiently fast will do. Once again, this is a weaker version adapted to\nour purpose.\nLemma 2. Let φ(α1, . . . , αr) be a quantifier-free first-order formula over the\nstructure (C, 0, 1, +, −, ×, =), containing only polynomials of degree less than\nD and whose coefficients are integers of absolute value strictly bounded by C.\nAssume furthermore that φ( ̄α) holds for all coefficients ̄α = (α1, . . . , αr) ∈Cr\nalgebraically independent over Q.\nThen φ( ̄β) holds for any sequence (β1, . . . , βr) of integers satisfying β1 ≥C\nand βj+1 ≥CDjβD\nj\n(for 1 ≤j ≤r −1).\nThe proof can be found in [11, Lemma 5.4] and relies on the lack of big\ninteger roots of multivariate polynomials.\nLet us sketch a first attempt to prove a constructive version of Lemma 1,\nnamely that n + 1 polynomials with integer coefficients are enough for defin-\ning V (this first try will not work but gives the idea of the proof of the next\nsection). The idea is to use Lemma 2 with the formula φ( ̄α) that tells us that\nthe n + 1 linear combinations of the fi with αi,j as coefficients define the same\nvariety as f1, . . . , fs. At first this formula is not quantifier-free, but over C we\ncan eliminate quantifiers while keeping degree and coefficients reasonably small\nthanks to Theorem 1. Lemma 1 asserts that φ( ̄α) holds as soon as the αi,j are\nalgebraically independent. Then Lemma 2 tells us that φ( ̄β) holds for integers\nβi,j growing fast enough. Thus V is now defined by n + 1 linear combinations of\nthe fi with integer coefficients.\nIn fact, this strategy fails to work for our purpose because the coefficients\ninvolved are growing too fast to be computed in polynomial space. That is why\nwe will proceed by stages in the proofs below: we adopt a divide-and-conquer\napproach and use induction.\n4.1\nTests of Membership\nThe base case of our induction is the following lemma, whose proof is sketched\nin the end of the preceding section. We only consider here a small number of\npolynomials, therefore avoiding the problem of too big coefficients mentioned in\nthe preceding section.\nLemma 3. There exists a polynomial q(n, d) such that, if V ⊆Cn is a variety\ndefined by 2(n + 1) polynomials f1, . . . , f2(n+1) ∈Z[x1, . . . , xn] of degree ≤2d\nand of coefficients bounded by 22M in absolute value, then:\n1. the variety V is defined by n + 1 polynomials g1, . . . , gn+1 ∈Z[x1, . . . , xn] of\ndegree ≤2d and of coefficients bounded by 22M+q(n,d) in absolute value;\n2. furthermore, the coefficients of the gi are bitwise computable from those of\nthe fj in working space Mq(n, d)."},{"paragraph_id":"p9","order":9,"text":"8\nPascal Koiran and Sylvain Perifel\nProof. The first-order formula φ( ̄α) (where ̄α ∈C2(n+1)2), expressing that the\nn + 1 linear combinations of the fj’s with coefficients ̄α also define V , can be\nwritten as follows:\nφ( ̄α) ≡∀x ∈Cn"},{"paragraph_id":"p10","order":10,"text":"n+1\n^\ni=1\n2(n+1)\nX\nj=1\nαi,jfj(x) = 0 ↔\n2(n+1)\n^\nj=1\nfj(x) = 0"},{"paragraph_id":"p11","order":11,"text":",\nwhere αi,j is a shorthand for α2(i−1)(n+1)+j. The polynomials in this formula are\nof degree ≤1 + 2d and their coefficients are bounded in absolute value by 22M .\nOver C, the quantifier of this formula can be eliminated by Theorem 1: φ( ̄α)\nis equivalent to a quantifier-free formula ψ( ̄α), the polynomials occuring in which\nhave their degree less than D = D(n, log(3(n + 1)), d + 1) and their coefficients\nstrictly bounded in absolute value by C = 22MD, where D(n, log(3(n + 1)), d +\n1) = 2p(n,log(3(n+1)),d+1) is defined in Theorem 1.\nBy Lemma 1, ψ( ̄α) holds for all coefficients ̄α algebraically independent, so\nthat we wish to apply Lemma 2 with integers βi growing sufficiently fast. Let\nr = (1 + 2(n + 1)2)p(n, log(3(n + 1)), d + 1), so that\nD ≤2r and CD2(n+1)2 ≤22M+r\nand define\nβi = 22M+2ir for 1 ≤i ≤2(n + 1)2.\nNote that for all i, βi ≤22M+4(n+1)2r, and it is furthermore easy to check that\nβ1 ≥C and βi+1 ≥CDiβD\ni . Thus by Lemma 2, ψ( ̄β) is true. Define the poly-\nnomial q(n, d) = 1 + 4(n + 1)2r (up to a multiplicative constant for the space\ncomplexity below). Now, letting\ngi =\n2(n+1)\nX\nj=1\nβi,jfj,\nwhere βi,j is a shorthand for β2(i−1)(n+1)+j, proves the first point of the lemma.\nFor the second point, remark that the coefficients βi are bitwise computable\nin space O(M + rn2) and that the coefficients of the gi are merely a sum of\n2(n + 1) products of βj and coefficients of the fk. This multiplication uses only\nspace O(M + rn2) since the integers involved have encoding size 2O(M+rn2) (in\nour case this is particularly easy because the βj are powers of 2). The 2n + 1\nadditions are also performed in space O(M + rn2). This proves the second point\nof the lemma.\n⊓⊔\nProposition 4 now follows by induction.\nProposition 4. There exists a polynomial p(n, s, d) such that, if V is a vari-\nety defined by 2s polynomials f1, . . . , f2s ∈Z[x1, . . . , xn] of degree ≤2d and of\ncoefficients bounded by 22M in absolute value, then:"},{"paragraph_id":"p12","order":12,"text":"VPSPACE and a transfer theorem over the complex field\n9\n1. the variety V is defined by n + 1 polynomials g1, . . . , gn+1 ∈Z[x1, . . . , xn] of\ndegree ≤2d and of coefficients bounded by 22M+p(n,s,d) in absolute value;\n2. moreover, the coefficients of the gi are bitwise computable from those of the\nfj in working space Mp(n, s, d).\nProof. This is done by induction on s. Take p(n, s, d) = sq(n, d) where q(n, d) is\nthe polynomial defined in Lemma 3. The base case 2s ≤2(n + 1) follows from\nLemma 3. Suppose therefore that 2s > 2(n + 1). Call V1 and V2 the varieties\ndefined respectively by f1, . . . , f2s−1 and by f2s−1+1, . . . , f2s. Then V = V1 ∩V2\nand by induction hypothesis, V1 and V2 are both defined by n+1 polynomials of\ndegree ≤2d whose coefficients are bounded by 22M+(s−1)q(n,d) in absolute value\nand computable in space M(s −1)q(n, d).\nTherefore by Lemma 3, V is defined by n + 1 polynomials of degree ≤2d\nwhose coefficients are bounded by 22M+sq(n,d) in absolute value and computable\nin space Msq(n, d) as claimed in the proposition.\n⊓⊔\n4.2\nUnion of Varieties\nIn our case, however, the tests made by the algorithm of Section 5 are not exactly\nof the form studied in the previous section: instead of a single variety given by\ns polynomials, we have to decide “x ∈W?” when W ⊆Cn is the union of k\nvarieties. Of course, since the union is finite W is also a variety, but the encoding\nis not the same as above: now, k sets of s polynomials are given.\nA first naive approach is to define W = ∪iVi by the different products of the\npolynomials defining the Vi, but it turns out that there are too many products to\nbe dealt with. Instead, we will adopt a divide-and-conquer scheme as previously.\nLemma 4. There exists a polynomial q(n, d) such that, if V1 and V2 are two\nvarieties of Cn, each defined by n + 1 polynomials in Z[x1, . . . , xn], respectively\nf1, . . . , fn+1 and g1, . . . , gn+1, of degree ≤2d and of coefficients bounded by 22M\nin absolute value, then:\n1. the variety V = V1 ∪V2 is defined by n + 1 polynomials h1, . . . , hn+1 in\nZ[x1, . . . , xn] of degree ≤2d+1 and of coefficients bounded by 22M+q(n,d) in\nabsolute value;\n2. the coefficients of the hi are bitwise computable from those of the fj and gk\nin space Mq(n, d).\nProof. The variety V is defined by the (n + 1)2 polynomials figj for 1 ≤i, j ≤\nn + 1: these polynomials have degree ≤2d+1. Note moreover that there are at\nmost 2n(d+1) monomials of fixed degree δ ≤2d+1, therefore the coefficients of\nthe figj are a sum of at most 2n(d+1) products of integers of encoding size 2M.\nThus they are computable in space O(Mnd) from those of the fi and gj. This\nalso shows that the coefficients of the products figj are bounded in absolute\nvalue by 2n(d+1)22M+1 ≤22M+1+n(d+1). Applying Proposition 4 now enables to\nconclude if we take q(n, d) = 1 + n(d + 1) + p(n, log((n + 1)2), d + 1), where p is\nthe polynomial defined in Proposition 4.\n⊓⊔"},{"paragraph_id":"p13","order":13,"text":"10\nPascal Koiran and Sylvain Perifel\nThe next proposition now follows by induction.\nProposition 5. There exists a polynomial r(n, s, k, d) such that, if V1, . . . , V2k ⊆\nCn are 2k varieties, Vi being defined by 2s polynomials f (i)\n1 , . . . , f (i)\n2s ∈Z[x1, . . . , xn]\nof degree ≤2d and of coefficients bounded by 22M in absolute value, then:\n1. the variety V\n= ∪2k\ni=1Vi is defined by n + 1 polynomials g1, . . . , gn+1 in\nZ[x1, . . . , xn] of degree ≤2d+k and whose coefficients are bounded in ab-\nsolute value by 22M+r(n,s,k,d);\n2. moreover, the coefficients of the gi are bitwise computable from those of the\nf (j)\nj′\nin space Mr(n, s, k, d).\nProof. We proceed by induction on k. Define r(n, s, k, d) = (k+1)(p(n, s, d+k)+\nq(n, d+k)), where p and q are defined in Proposition 4 and Lemma 4 respectively.\nThe base case k = 0 is merely an application of Proposition 4. For k > 0, we first\napply Proposition 4 to the Vi, so that each variety Vi is now defined by n + 1\npolynomials of degree ≤2d and whose coefficients are bounded in absolute value\nby 22M+p(n,s,d) and computable in space Mp(n, s, d). Let us group the varieties\nVi by pairs: call Wi = V2i−1 ∪V2i for 1 ≤i ≤2k−1. There are 2k−1 varieties Wi\nand we have V = ∪iWi. By Lemma 4, each variety Wi is defined by n + 1 poly-\nnomials of degree ≤2d+1, of coefficients of bitsize 2M+p(n,s,d)+q(n,d) and bitwise\ncomputable in space M(p(n, s, d) + q(n, d)). By induction hypothesis at rank\nk −1, V is defined by n+1 polynomials of degree ≤2d+1+(k−1), of coefficients of\nbitsize 2M+p(n,s,d)+q(n,d)+k(p(n,⌈log(n+1)⌉,d+k−1)+q(n,d+k−1)) ≤2M+r(n,s,k,d) and\nbitwise computable in space Mr(n, s, k, d). This proves the proposition.\n⊓⊔\nHere is the main consequence on membership tests to a union of varieties.\nCorollary 1. Let p(n) and q(n) be two polynomials. Suppose (fn( ̄x, ̄y, ̄z)) is a\nUniform VPSPACE0 family with | ̄x| = n, | ̄y| = p(n) and | ̄z| = q(n). For an integer\n0 ≤i < 2p(n), call V (n)\ni\n⊆Cn the variety defined by the polynomials fn( ̄x, i, j)\nfor 0 ≤j < 2q(n) (in this notation, i and j are encoded in binary).\nThen there exists a Uniform VPSPACE0 family gn( ̄x, ̄y, ̄z), where | ̄x| = n,\n| ̄y| = p(n) and | ̄z| = ⌈log(n + 1)⌉, such that\n∀ ̄x ∈Cn,\n∀k < 2p(n),"},{"paragraph_id":"p14","order":14,"text":"̄x ∈\nk[\ni=0\nV (n)\ni\n⇐⇒\nn\n^\nj=0\ngn( ̄x, k, j) = 0"},{"paragraph_id":"p15","order":15,"text":".\nProof. If (fn) is a Uniform VPSPACE0 family, by definition there exists a poly-\nnomial p(n) such that the degree of fn is bounded by 2p(n) and the absolute\nvalue of the coefficients by 22p(n). Therefore d, M, s and k are polynomially\nbounded in Proposition 5 and the space needed to compute the coefficients of\ngn is polynomial.\n⊓⊔"},{"paragraph_id":"p16","order":16,"text":"VPSPACE and a transfer theorem over the complex field\n11\n5\nProof of the Main Theorem\nSign conditions are the main ingredient of the proof. Over C, we define the\n“sign” of a ∈C by 0 if a = 0 and 1 otherwise. Let us fix a family of polynomials\nf1, . . . , fs ∈Z[x1, . . . , xn]. A sign condition is an element S ∈{0, 1}s. Hence\nthere are 2s sign conditions. Intuitively, the i-th component of a sign condition\ndetermines the sign of the polynomial fi.\n5.1\nSatisfiable Sign Conditions\nThe sign condition of a point ̄x ∈Cn is the tuple S ̄x ∈{0, 1}s defined by\nS ̄x\ni = 0 ⇐⇒fi( ̄x) = 0. We say that a sign condition is satisfiable if it is the\nsign condition of some ̄x ∈Cn. As 0-1 tuples, sign conditions can be viewed as\nsubsets of {1, . . ., s}. Using a fast parallel sorting algorithm (e.g. Cole’s, [7]),\nwe can sort satisfiable sign conditions in polylogarithmic parallel time in a way\ncompatible with set inclusion (e.g. the lexicographic order). We now fix such\na compatible linear order on sign conditions and consider our satisfiable sign\nconditions S(1) < S(2) < . . . < S(N) sorted accordingly.\nThe key point resides in the following theorem, coming from the algorithm\nof [9]: there is a “small” number of satisfiable sign conditions and enumerating\nthem is “easy”.\nTheorem 3. Let f1, . . . , fs ∈Z[x1, . . . , xn] and d be their maximal degree. Then\nthe number of satisfiable sign conditions is N = (sd)O(n), and there is a uniform\nalgorithm working in space\n n log(sd)\n O(1) which, on boolean input f1, . . . , fs (in\ndense representation) and (i, j) in binary, returns the j-th component of the i-th\nsatisfiable sign condition.\nWhen log(sd) is polynomial in n, as will be the case, this yields a PSPACE\nalgorithm. If furthermore the coefficients of fi are computable in polynomial\nspace, we will then be able to use the satisfiable sign conditions in the coefficients\nof VPSPACE families, as in Lemma 5 below.\nLet us explain why we are interested in sign conditions. An arithmetic circuit\nperforms tests of the form f( ̄x) = 0 on input ̄x ∈Cn, where f is a polynomial.\nSuppose f1, . . . , fs is the list of all polynomials that can be tested in any possible\ncomputation. Then two elements of Cn with the same sign condition are simul-\ntaneously accepted or rejected by the circuit: the results of the tests are indeed\nalways the same for both elements.\nThus, instead of finding out whether ̄x ∈Cn is accepted by the circuit, it is\nenough to find out whether the sign condition of ̄x is accepted. The advantage\nresides in handling only boolean tuples (the sign conditions) instead of complex\nnumbers (the input ̄x). But we have to be able to find the sign condition of\nthe input ̄x. This requires first the enumeration of all the polynomials possibly\ntested in any computation of the circuit."},{"paragraph_id":"p17","order":17,"text":"12\nPascal Koiran and Sylvain Perifel\n5.2\nEnumerating all Possibly Tested Polynomials\nIn the execution of an algebraic circuit, the values of some polynomials at the\ninput ̄x are tested to zero. In order to find the sign condition of the input ̄x, we\nhave to be able to enumerate in polynomial space all the polynomials that can\never be tested to zero in the computations of an algebraic circuit. This is done\nlevel by level as in [8, Th. 3] and [14].\nProposition 6. Let C be a constant-free algebraic circuit with n variables and\nof depth d.\n1. The number of different polynomials possibly tested to zero in the computa-\ntions of C is 2d2O(n).\n2. There exists an algorithm using work space (nd)O(1) which, on input C and\nintegers (i, j) in binary, outputs the j-th bit of the representation of the i-th\npolynomial.\nTogether with Theorem 3, this enables us to prove the following result which\nwill be useful in the proof of Proposition 7: in Uniform VPSPACE0 we can enu-\nmerate the polynomials as well as the satisfiable sign conditions.\nLemma 5. Let (Cn) be a uniform family of polynomial-depth algebraic circuits\nwith polynomially many inputs. Call d(n) the depth of Cn and i(n) the number\nof inputs. Let f (n)\n1\n, . . . , f (n)\ns\nbe all the polynomials possibly tested to zero by Cn\nas in Proposition 6, where s = 2O(nd(n)2). There are therefore N = 2O(n2d(n)2)\nsatisfiable sign conditions S(1), . . . , S(N) by Theorem 3.\nThen there exists a Uniform VPSPACE0 family (gn( ̄x, ̄y, ̄z)), where | ̄x| = i(n),\n| ̄y| = O(n2d(n)2) and | ̄z| = O(nd(n)2), such that for all 1 ≤i ≤N and 1 ≤j ≤\ns, we have:\ngn( ̄x, i, j) =\n(\n0\nif S(i)\nj\n= 1\nf (n)\nj\n( ̄x) otherwise.\n5.3\nFinding the Sign Condition of the Input\nIn order to find the sign condition S ̄x of the input ̄x ∈Cn, we will give a\npolynomial-time algorithm which tests some VPSPACE family for zero. Here is\nthe formalized notion of a polynomial-time algorithm with VPSPACE tests.\nDefinition 3. A polynomial-time algorithm with Uniform VPSPACE0 tests is\na Uniform VPSPACE0 family (fn(x1, . . . , xu(n))) together with a uniform fam-\nily (Cn) of constant-free polynomial-size algebraic circuits endowed with spe-\ncial test gates of indegree u(n), whose value is 1 on input (a1, . . . , au(n)) if\nfn(a1, . . . , au(n)) = 0 and 0 otherwise.\nObserve that a constant number of Uniform VPSPACE0 families can be used in\nthe preceding definition instead of only one: it is enough to combine them all in\none by using “selection variables”."},{"paragraph_id":"p18","order":18,"text":"VPSPACE and a transfer theorem over the complex field\n13\nThe precise result we show now is the following. By the “rank” of a satisfiable\nsign condition, we merely mean its index in the fixed order on satisfiable sign\nconditions.\nProposition 7. Let (Cn) be a uniform family of algebraic circuits of polynomial\ndepth and with a polynomial number i(n) of inputs. There exists a polynomial-\ntime algorithm with Uniform VPSPACE0 tests which, on input ̄x ∈Ci(n), returns\nthe rank i of the sign condition S(i) of ̄x with respect to the polynomials g1, . . . , gs\ntested to zero by Cn given by Proposition 6.\nProof. Take the Uniform VPSPACE0 family (gn( ̄x, ̄y, ̄z)) as in Lemma 5: in essence,\ngn enumerates all the polynomials f1, . . . , fs possibly tested to zero in Cn and\nenumerates the N satisfiable sign conditions S(1) < . . . < S(N). The idea now is\nto perform a binary search in order to find the rank i of the sign condition of\nthe input ̄x.\nLet S(j) ∈{0, 1}s be a satisfiable sign condition. We say that S(j) is a candi-\ndate whenever ∀m ≤s, S(j)\nm = 0 ⇒fm( ̄x) = 0. Remark that the sign condition\nof ̄x is the smallest candidate. Call Vj the variety defined by the polynomi-\nals {fm|S(j)\nm\n= 0}: by definition of gn, Vj is also defined by the polynomials\ngn( ̄x, j, k) for k = 1 to s. Note that S(j) is a candidate if and only if ̄x ∈Vj.\nCorollary 1 combined with Lemma 5 asserts that tests of the form ̄x ∈∪k≤jVk\nare in Uniform VPSPACE0. They are used to perform a binary search by making\nj vary. In a number of steps logarithmic in N (i.e. polynomial in n), we find the\nrank i of the sign condition of ̄x.\n⊓⊔\n5.4\nA Polynomial-time Algorithm for PARC Problems\nLemma 6. Let (Cn) be a uniform family of constant-free polynomial-depth al-\ngebraic circuits. There is a (boolean) algorithm using work space polynomial in\nn which, on input i, decides whether the elements of the i-th satisfiable sign\ncondition S(i) are accepted by the circuit Cn.\nProof. We follow the circuit Cn level by level. For test gates, we compute the\npolynomial f to be tested. Then we enumerate the polynomials f1, . . . , fs as\nin Proposition 6 for the circuit Cn and we find the index j of f in this list.\nBy consulting the j-th bit of the i-th satisfiable sign condition with respect to\nf1, . . . , fs (which is done by the polynomial-space algorithm of Theorem 3), we\ntherefore know the result of the test and can go on like this until the output\ngate.\n⊓⊔\nTheorem 4. Let A ∈PAR0\nC. There exists a polynomial-time algorithm with\nUniform VPSPACE0 tests that decides A.\nProof. A is decided by a uniform family (Cn) of constant-free polynomial-depth\nalgebraic circuits. On input ̄x, thanks to Proposition 7 we first find the rank i of\nthe sign condition of ̄x with respect to the polynomials f1, . . . , fs of Proposition 6.\nThen we conclude by a last Uniform VPSPACE0 test simulating the polynomial-\nspace algorithm of Lemma 6 on input i.\n⊓⊔"},{"paragraph_id":"p19","order":19,"text":"14\nPascal Koiran and Sylvain Perifel\nTheorem 2 follows immediately from this result. One could obtain other ver-\nsions of these two results by changing the uniformity conditions or the role of\nconstants.\nReferences\n1. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation.\nSpringer-Verlag, 1998.\n2. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over\nthe real numbers: NP-completeness, recursive functions and universal machines.\nBulletin of the American Mathematical Society, 21(1):1–46, 1989.\n3. P. B ̈urgisser. Completeness and Reduction in Algebraic Complexity Theory, vol-\nume 7 of Algorithms and Computation in Mathematics. Springer, 2000.\n4. P. B ̈urgisser. On implications between P-NP-hypotheses: Decision versus compu-\ntation in algebraic complexity. In Mathematical Foundations of Computer Science,\nvolume 2316 of Lecture Notes in Computer Science, pages 3–17. Springer, 2001.\n5. O. Chapuis and P. Koiran. Saturation and stability in the theory of computation\nover the reals. Annals of Pure and Applied Logic, 99:1–49, 1999.\n6. P. Charbit, E. Jeandel, P. Koiran, S. Perifel, and S. Thomass ́e. Finding a vector\northogonal to roughly half a collection of vectors. Available from http://perso.ens-\nlyon.fr/pascal.koiran/publications.html. Accepted for publication in Journal of\nComplexity, 2006.\n7. R. Cole. Parallel merge sort. SIAM J. Comput., 17(4):770–785, 1988.\n8. F. Cucker and D. Grigoriev. On the power of real Turing machines over binary\ninputs. SIAM Journal on Computing, 26(1):243–254, 1997.\n9. N. Fitchas, A. Galligo, and J. Morgenstern. Precise sequential and parallel com-\nplexity bounds for quantifier elimination over algebraically closed fields. Journal\nof Pure and Applied Algebra, 67:1–14, 1990.\n10. D. Grigoriev. Topological complexity of the range searching. Journal of Complexity,\n16:50–53, 2000.\n11. P. Koiran. Randomized and deterministic algorithms for the dimension of algebraic\nvarieties. In Proc. 38th IEEE Symposium on Foundations of Computer Science,\npages 36–45, 1997.\n12. P. Koiran. Circuits versus trees in algebraic complexity. In Proc. STACS 2000,\nvolume 1770 of Lecture Notes in Computer Science, pages 35–52. Springer, 2000.\n13. P. Koiran and S. Perifel.\nValiant’s model: from exponential sums to exponen-\ntial products. In Mathematical Foundations of Computer Science, volume 4162 of\nLecture Notes in Computer Science, pages 596–607. Springer-Verlag, 2006.\n14. P. Koiran and S. Perifel. VPSPACE and a transfer theorem over the reals. In\nProc. 24th Symposium on Theoretical Aspects of Computer Science, volume 4393\nof Lecture Notes in Computer Science, pages 417–428, 2007. Long version available\nfrom http://perso.ens-lyon.fr/pascal.koiran/publications.html.\n15. G. Malod. Polynˆomes et coefficients. PhD thesis, Universit ́e Claude Bernard Lyon\n1, July 2003. Available from http://tel.archives-ouvertes.fr/tel-00087399.\n16. G. Malod and N. Portier. Characterizing Valiant’s algebraic complexity classes.\nIn Mathematical Foundations of Computer Science, volume 4162 of Lecture Notes\nin Computer Science, pages 704–716. Springer-Verlag, 2006.\n17. B. Poizat. Les petits cailloux. Al ́eas, 1995.\n18. L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on\nTheory of Computing, pages 249–261, 1979."}],"pages":[{"page":1,"text":"arXiv:0706.1477v1 [cs.CC] 11 Jun 2007\nVPSPACE and a transfer theorem over the\ncomplex field\nPascal Koiran and Sylvain Perifel\nLIP⋆, ́Ecole Normale Sup ́erieure de Lyon.\n[Pascal.Koiran,Sylvain.Perifel]@ens-lyon.fr\nJune 2007\nAbstract. We extend the transfer theorem of [14] to the complex field. That is,\nwe investigate the links between the class VPSPACE of families of polynomials\nand the Blum-Shub-Smale model of computation over C. Roughly speaking,\na family of polynomials is in VPSPACE if its coefficients can be computed in\npolynomial space. Our main result is that if (uniform, constant-free) VPSPACE\nfamilies can be evaluated efficiently then the class PARC of decision problems\nthat can be solved in parallel polynomial time over the complex field collapses\nto PC. As a result, one must first be able to show that there are VPSPACE\nfamilies which are hard to evaluate in order to separate PC from NPC, or even\nfrom PARC.\nKeywords: computational complexity, algebraic complexity, Blum-Shub-Smale\nmodel, Valiant’s model.\n1\nIntroduction\nIn algebraic complexity theory, two main categories of problems are studied:\nevaluation and decision problems. The evaluation of the permanent of a matrix\nis a typical example of an evaluation problem, and it is well known that the\npermanent family is complete for the class VNP of “easily definable” polynomial\nfamilies [18]. Deciding whether a system of polynomial equations has a solution\nover C is a typical example of a decision problem. This problem is NP-complete\nin the Blum-Shub-Smale model of computation over the complex field [1,2].\nThe main purpose of this paper is to provide a transfer theorem connecting\nthe complexity of evaluation and decision problems. This paper is therefore in\nthe same spirit as [13] and [14] (see also [4]). In the present paper we work with\nthe class of polynomial families VPSPACE introduced in [14]. Roughly speaking,\na family of polynomials (of possibly exponential degree) is in VPSPACE if its\ncoefficients can be evaluated in polynomial space. For instance, it is shown in [14]\nthat resultants of systems of multivariate polynomial equations form a VPSPACE\nfamily. The main result in [14] was that if (uniform, constant-free) VPSPACE\nfamilies can be evaluated efficiently then the class PARR of decision problems\nthat can be solved in parallel polynomial time over the real numbers collapses\nto PR.\n⋆UMR 5668 ENS Lyon, CNRS, UCBL, INRIA. Research report RR2007-27."},{"page":2,"text":"2\nPascal Koiran and Sylvain Perifel\nHere we extend this result to the complex field C. At first glance the result\nseems easier because the order ≤over the reals does not have to be taken into\naccount. The result of [14] indeed makes use of a clever combinatorial lemma\nof [10] on the existence of a vector orthogonal to roughly half a collection of\nvectors. More precisely, it relies on the constructive version of this lemma [6].\nOn the complex field, we do not need this construction.\nBut the lack of an order over C makes another part of the proof more difficult.\nIndeed, over R testing whether a point belongs to a real variety is done by testing\nwhether the sum of the squares of the polynomials is zero, a trick that cannot be\nused over the complex field. Hence one of the main technical developments of this\npaper is to explain how to decide with a small number of tests whether a point\nis in the complex variety defined by an exponential number of polynomials. This\nenables us to follow the nonconstructive proof of [12] for our transfer theorem.\nTherefore, the main result of the present paper is that if (uniform, constant-\nfree) VPSPACE families can be evaluated efficiently then the class PARC of deci-\nsion problems that can be solved in parallel polynomial time over the complex\nfield collapses to PC (this is precisely stated in Theorem 2). The class PARC plays\nroughly the same role in the theory of computation over the complex field as\nPSPACE in discrete complexity theory. In particular, it contains NPC [1] (but the\nproof of this inclusion is much more involved than in the discrete case). It follows\nfrom our main result that in order to separate PC from NPC, or even from PARC,\none must first be able to show that there are VPSPACE families which are hard\nto evaluate. This seems to be a very challenging lower bound problem, but it is\nstill presumably easier than showing that the permanent is hard to evaluate.\nOrganization of the paper. We first recall in Section 2 some usual notions\nand notations concerning algebraic complexity (Valiant’s model, the Blum-Shub-\nSmale model) and quantifier elimination. The class VPSPACE is defined in Sec-\ntion 3 and some properties proved in [14] are given. Section 4 explains how to\ndecide with a polynomial number of VPSPACE tests whether a point belongs to\na variety. The main difficulty here is that the variety is given as a union of an\nexponential number of varieties, each defined by an exponential number of poly-\nnomials. Finally, Section 5 is devoted to the proof of the transfer theorem. Sign\nconditions are the main tool in this section. We show that PARC problems are\ndecided in polynomial time if we allow Uniform VPSPACE0 tests. The transfer\ntheorem follows as a corollary.\n2\nNotations and Preliminaries\n2.1\nThe Blum-Shub-Smale Model\nIn contrast with boolean complexity, algebraic complexity deals with other struc-\ntures than {0, 1}. In this paper we will focus on the complex field (C, +, −, ×, =).\nAlthough the original definitions of Blum, Shub and Smale [2,1] are in terms of\nuniform machines, we will follow [17] by using families of algebraic circuits to\nrecognize languages over C, that is, subsets of C∞= S\nn≥0 Cn."},{"page":3,"text":"VPSPACE and a transfer theorem over the complex field\n3\nAn algebraic circuit is a directed acyclic graph whose vertices, called gates,\nhave indegree 0, 1 or 2. An input gate is a vertex of indegree 0. An output gate is\na gate of outdegree 0. We assume that there is only one such gate in the circuit.\nGates of indegree 2 are labelled by a symbol from the set {+, −, ×}. Gates of\nindegree 1, called test gates, are labelled “= 0?”. The size of a circuit C, in\nsymbols |C|, is the number of vertices of the graph.\nA circuit with n input gates computes a function from Cn to C. On input\n ̄u ∈Cn the value returned by the circuit is by definition equal to the value of\nits output gate. The value of a gate is defined in the usual way. Namely, the\nvalue of input gate number i is equal to the i-th input ui. The value of other\ngates is then defined recursively: it is the sum of the values of its entries for a\n+-gate, their difference for a −-gate, their product for a ×-gate. The value taken\nby a test gate is 0 if the value of its entry is ̸= 0 and 1 otherwise. Since we are\ninterested in decision problems, we assume that the output is a test gate: the\nvalue returned by the circuit is therefore 0 or 1.\nThe class PC is the set of languages L ⊆C∞such that there exists a tuple\n ̄a ∈Cp and a P-uniform family of polynomial-size circuits (Cn) satisfying the\nfollowing condition: Cn has exactly n + p inputs, and for any ̄x ∈Cn, ̄x ∈L ⇔\nCn( ̄x, ̄a) = 1. The P-uniformity condition means that Cn can be built in time\npolynomial in n by an ordinary (discrete) Turing machine. Note that ̄a plays the\nrole of the machine constants of [1,2].\nAs in [5], we define the class PARC as the set of languages over C recognized\nby a PSPACE-uniform (or equivalently P-uniform) family of algebraic circuits of\npolynomial depth (and possibly exponential size), with constants ̄a as for PC.\nNote at last that we could also define similar classes without constants ̄a. We\nwill use the superscript 0 to denote these constant-free classes, for instance P0\nC\nand PAR0\nC.\nWe end this section with a theorem on the first-order theory of the complex\nnumbers: quantifiers can be eliminated without much increase of the coefficients\nand degree of the polynomials. We give a weak version of the result of [9]:\nin particular, we do not need efficient elimination algorithms. Note that the\nonly allowed constants in our formulae are 0 and 1 (in particular, only integer\ncoefficients can appear). For notational consistency with the remainding of the\npaper, we denote by 2s, 2d and 22M the number of polynomials, their degree\nand the absolute value of their coefficients respectively. This will simplify the\ncalculations and emphasize that s, d and M will be polynomial. Note furthermore\nthat the polynomial p(n, s, d) in the theorem is independent of the formula φ.\nTheorem 1. Let φ be a first-order formula over (C, 0, 1, +, −, ×, =) of the form\n∀ ̄xψ( ̄x), where ̄x is a tuple of n variables and ψ a quantifier-free formula where\n2s polynomials occur. Suppose that their degrees are bounded by 2d and their\ncoefficients by 22M in absolute value.\nThere exists a polynomial p(n, s, d), independent of φ, such that the formula\nφ is equivalent to a quantifier-free formula ψ in which all polynomials have degree\nless than D(n, s, d) = 2p(n,s,d), and their coefficients are integers strictly bounded\nin absolute value by 22MD(n,s,d)."},{"page":4,"text":"4\nPascal Koiran and Sylvain Perifel\n2.2\nValiant’s Model\nIn Valiant’s model, one computes polynomials instead of recognizing languages.\nWe thus use arithmetic circuits instead of algebraic circuits. A book-length treat-\nment of this topic can be found in [3].\nAn arithmetic circuit is the same as an algebraic circuit but test gates are not\nallowed. That is to say we have indeterminates x1, . . . , xu(n) as input together\nwith arbitrary constants of C; there are +, −and ×-gates, and we therefore\ncompute multivariate polynomials.\nThe polynomial computed by an arithmetic circuit is defined in the usual way\nby the polynomial computed by its output gate. Thus a family (Cn) of arithmetic\ncircuits computes a family (fn) of polynomials, fn ∈C[x1, . . . , xu(n)]. The class\nVPnb defined in [15] is the set of families (fn) of polynomials computed by a\nfamily (Cn) of polynomial-size arithmetic circuits, i.e., Cn computes fn and there\nexists a polynomial p(n) such that |Cn| ≤p(n) for all n. We will assume without\nloss of generality that the number u(n) of variables is bounded by a polynomial\nfunction of n. The subscript nb indicates that there is no bound on the degree\nof the polynomial, in contrast with the original class VP of Valiant where a\npolynomial bound on the degree of the polynomial computed by the circuit is\nrequired. Note that these definitions are nonuniform. The class Uniform VPnb\nis obtained by adding a condition of polynomial-time uniformity on the circuit\nfamily, as in Section 2.1.\nWe can also forbid constants from our arithmetic circuits in unbounded-\ndegree classes, and define constant-free classes. The only constant allowed is\n1 (in order to allow the computation of constant polynomials). As for classes\nof decision problems, we will use the superscript 0 to indicate the absence of\nconstant: for instance, we will write VP0\nnb (for bounded-degree classes, we are to\nbe more careful: the “formal degree” of the circuits comes into play, see [15,16]).\n3\nThe Class VPSPACE\nThe class VPSPACE was introduced in [14]. Some of its properties are given there\nand a natural example of a VPSPACE family coming from algebraic geometry,\nnamely the resultant of a system of polynomial equations, is provided. In this\nsection, after the definition we give some properties without proof and refer\nto [14] for further details.\n3.1\nDefinition\nWe fix an arbitrary field K. The definition of VPSPACE will be stated in terms\nof coefficient function. A monomial xα1\n1 · · · xαn\nn\nis encoded in binary by α =\n(α1, . . . , αn) and will be written ̄xα.\nDefinition 1. Let (fn) be a family of multivariate polynomials with integer co-\nefficients. The coefficient function of (fn) is the function a whose value on input"},{"page":5,"text":"VPSPACE and a transfer theorem over the complex field\n5\n(n, α, i) is the i-th bit a(n, α, i) of the coefficient of the monomial ̄xα in fn. Fur-\nthermore, a(n, α, 0) is the sign of the coefficient of the monomial ̄xα. Thus fn\ncan be written as\nfn( ̄x) =\nX\nα\n \n(−1)a(n,α,0) X\ni≥1\na(n, α, i)2i−1 ̄xα \n.\nThe coefficient function is a function a : {0, 1}∗→{0, 1} and can therefore be\nviewed as a language. This allows us to speak of the complexity of the coefficient\nfunction. Note that if K is of characteristic p > 0, then the coefficients of our\npolynomials will be integers modulo p (hence with a constant number of bits).\nIn this paper, we will focus only on the field C (which is of characteristic 0).\nDefinition 2. The class Uniform VPSPACE0 is the set of all families (fn) of\nmultivariate polynomials fn ∈K[x1, . . . , xu(n)] satisfying the following require-\nments:\n1. the number u(n) of variables is polynomially bounded;\n2. the polynomials fn have integer coefficients;\n3. the size of the coefficients of fn is bounded by 2p(n) for some polynomial p;\n4. the degree of fn is bounded by 2p(n) for some polynomial p;\n5. the coefficient function of (fn) is in PSPACE.\nWe have chosen to present only Uniform VPSPACE0, a uniform class without\nconstants, because this is the main object of study in this paper. In keeping\nwith the tradition set by Valiant, however, the class VPSPACE is nonuniform\nand allows for arbitrary constants. See [14] for a precise definition.\n3.2\nAn Alternative Characterization and Some Properties\nLet Uniform VPAR0 be the class of families of polynomials computed by a\nPSPACE-uniform family of constant-free arithmetic circuits of polynomial depth\n(and possibly exponential size). This in fact characterizes Uniform VPSPACE0.\nThe proof is given in [14].\nProposition 1. The two classes Uniform VPSPACE0 and Uniform VPAR0 are\nequal.\nWe see here the similarity with PARC, which by definition are those languages\nrecognized by uniform algebraic circuits of polynomial depth. But of course there\nis no test gate in the arithmetic circuits of Uniform VPAR0.\nWe now turn to some properties of VPSPACE. The following two propositions\ncome from [14]. They stress the unlikeliness of the hypothesis that VPSPACE has\npolynomial-size circuits.\nProposition 2. Assuming the generalized Riemann hypothesis (GRH), VPnb =\nVPSPACE if and only if [P/poly = PSPACE/poly and VP = VNP]."},{"page":6,"text":"6\nPascal Koiran and Sylvain Perifel\nProposition 3. Uniform VPSPACE0 = Uniform VP0\nnb =⇒PSPACE = P-uniform NC.\nRemark 1. To the authors’ knowledge, the separation “PSPACE ̸= P-uniform NC”\nis not known to hold (by contrast, PSPACE can be separated from logspace-\nuniform NC thanks to the space hierarchy theorem).\nLet us now state the main result of this paper.\nTheorem 2 (main theorem). If Uniform VPSPACE0 = Uniform VP0\nnb then\nPAR0\nC = P0\nC.\nNote that the collapse of the constant-free class PAR0\nC to P0\nC implies PARC =\nPC: just replace constants by new variables so as to transform a PARC problem\ninto a PAR0\nC problem, and then replace these variables by their original values\nso as to transform a P0\nC problem into a PC problem.\nThe next section is devoted to the problem of testing whether a point be-\nlongs to a variety. This problem is useful for the proof of the theorem: indeed,\nfollowing [12], several tests of membership to a variety will be made; the point\nhere is to make them constructive and efficient. The main difficulty is that the\nvariety can be defined by an exponential number of polynomials.\n4\nTesting Membership to a Union of Varieties\nIn this section we explain how to perform in Uniform VPSPACE0 membership\ntests of the form “ ̄x ∈V ”, where V ⊆Cn is a variety. We begin in Section 4.1 by\nthe case where V is given by s polynomials. In that case, we determine after some\nprecomputation whether ̄x ∈V in n + 1 tests. We first need two lemmas given\nbelow in order to reduce the number of polynomials and to replace transcendental\nelements by integers.\nThen, in Section 4.2, we deal with the case where V is given as a union of\nan exponential number of such varieties, as in the actual tests of the algorithm\nof Section 5. Determining whether ̄x ∈V still requires n + 1 tests, but the\nprecomputation is slightly heavier.\nLet us first state two useful lemmas. Suppose a variety V is defined by\nf1, . . . , fs, where fi ∈Z[x1, . . . , xn]. We are to determine whether ̄x ∈V with\nonly n + 1 tests, however big s might be. In a nonconstructive manner, this is\npossible and relies on the following classical lemma already used (and proved)\nin [12]: any n + 1 “generic” linear combinations of the fi also define V (the\nresult holds over any infinite field but here we need it only over C). We state\nthis lemma explicitly since we will also need it in our constructive proof.\nLemma 1. Let f1, . . . , fs ∈Z[x1, . . . , xn] be polynomials and V be the variety\nof Cn they define. Then for all coefficients (αi,j)i=1..s,j=1..n+1 ∈Cs(n+1) alge-\nbraically independent over Q, the n + 1 linear combinations gj = Ps\ni=1 αi,jfi\n(for j from 1 to n + 1) also define V ."},{"page":7,"text":"VPSPACE and a transfer theorem over the complex field\n7\nUnfortunately, in our case we cannot use transcendental numbers and must\nreplace them by integers. The following lemma from [11] asserts that integers\ngrowing sufficiently fast will do. Once again, this is a weaker version adapted to\nour purpose.\nLemma 2. Let φ(α1, . . . , αr) be a quantifier-free first-order formula over the\nstructure (C, 0, 1, +, −, ×, =), containing only polynomials of degree less than\nD and whose coefficients are integers of absolute value strictly bounded by C.\nAssume furthermore that φ( ̄α) holds for all coefficients ̄α = (α1, . . . , αr) ∈Cr\nalgebraically independent over Q.\nThen φ( ̄β) holds for any sequence (β1, . . . , βr) of integers satisfying β1 ≥C\nand βj+1 ≥CDjβD\nj\n(for 1 ≤j ≤r −1).\nThe proof can be found in [11, Lemma 5.4] and relies on the lack of big\ninteger roots of multivariate polynomials.\nLet us sketch a first attempt to prove a constructive version of Lemma 1,\nnamely that n + 1 polynomials with integer coefficients are enough for defin-\ning V (this first try will not work but gives the idea of the proof of the next\nsection). The idea is to use Lemma 2 with the formula φ( ̄α) that tells us that\nthe n + 1 linear combinations of the fi with αi,j as coefficients define the same\nvariety as f1, . . . , fs. At first this formula is not quantifier-free, but over C we\ncan eliminate quantifiers while keeping degree and coefficients reasonably small\nthanks to Theorem 1. Lemma 1 asserts that φ( ̄α) holds as soon as the αi,j are\nalgebraically independent. Then Lemma 2 tells us that φ( ̄β) holds for integers\nβi,j growing fast enough. Thus V is now defined by n + 1 linear combinations of\nthe fi with integer coefficients.\nIn fact, this strategy fails to work for our purpose because the coefficients\ninvolved are growing too fast to be computed in polynomial space. That is why\nwe will proceed by stages in the proofs below: we adopt a divide-and-conquer\napproach and use induction.\n4.1\nTests of Membership\nThe base case of our induction is the following lemma, whose proof is sketched\nin the end of the preceding section. We only consider here a small number of\npolynomials, therefore avoiding the problem of too big coefficients mentioned in\nthe preceding section.\nLemma 3. There exists a polynomial q(n, d) such that, if V ⊆Cn is a variety\ndefined by 2(n + 1) polynomials f1, . . . , f2(n+1) ∈Z[x1, . . . , xn] of degree ≤2d\nand of coefficients bounded by 22M in absolute value, then:\n1. the variety V is defined by n + 1 polynomials g1, . . . , gn+1 ∈Z[x1, . . . , xn] of\ndegree ≤2d and of coefficients bounded by 22M+q(n,d) in absolute value;\n2. furthermore, the coefficients of the gi are bitwise computable from those of\nthe fj in working space Mq(n, d)."},{"page":8,"text":"8\nPascal Koiran and Sylvain Perifel\nProof. The first-order formula φ( ̄α) (where ̄α ∈C2(n+1)2), expressing that the\nn + 1 linear combinations of the fj’s with coefficients ̄α also define V , can be\nwritten as follows:\nφ( ̄α) ≡∀x ∈Cn\n \n \nn+1\n^\ni=1\n2(n+1)\nX\nj=1\nαi,jfj(x) = 0 ↔\n2(n+1)\n^\nj=1\nfj(x) = 0\n \n ,\nwhere αi,j is a shorthand for α2(i−1)(n+1)+j. The polynomials in this formula are\nof degree ≤1 + 2d and their coefficients are bounded in absolute value by 22M .\nOver C, the quantifier of this formula can be eliminated by Theorem 1: φ( ̄α)\nis equivalent to a quantifier-free formula ψ( ̄α), the polynomials occuring in which\nhave their degree less than D = D(n, log(3(n + 1)), d + 1) and their coefficients\nstrictly bounded in absolute value by C = 22MD, where D(n, log(3(n + 1)), d +\n1) = 2p(n,log(3(n+1)),d+1) is defined in Theorem 1.\nBy Lemma 1, ψ( ̄α) holds for all coefficients ̄α algebraically independent, so\nthat we wish to apply Lemma 2 with integers βi growing sufficiently fast. Let\nr = (1 + 2(n + 1)2)p(n, log(3(n + 1)), d + 1), so that\nD ≤2r and CD2(n+1)2 ≤22M+r\nand define\nβi = 22M+2ir for 1 ≤i ≤2(n + 1)2.\nNote that for all i, βi ≤22M+4(n+1)2r, and it is furthermore easy to check that\nβ1 ≥C and βi+1 ≥CDiβD\ni . Thus by Lemma 2, ψ( ̄β) is true. Define the poly-\nnomial q(n, d) = 1 + 4(n + 1)2r (up to a multiplicative constant for the space\ncomplexity below). Now, letting\ngi =\n2(n+1)\nX\nj=1\nβi,jfj,\nwhere βi,j is a shorthand for β2(i−1)(n+1)+j, proves the first point of the lemma.\nFor the second point, remark that the coefficients βi are bitwise computable\nin space O(M + rn2) and that the coefficients of the gi are merely a sum of\n2(n + 1) products of βj and coefficients of the fk. This multiplication uses only\nspace O(M + rn2) since the integers involved have encoding size 2O(M+rn2) (in\nour case this is particularly easy because the βj are powers of 2). The 2n + 1\nadditions are also performed in space O(M + rn2). This proves the second point\nof the lemma.\n⊓⊔\nProposition 4 now follows by induction.\nProposition 4. There exists a polynomial p(n, s, d) such that, if V is a vari-\nety defined by 2s polynomials f1, . . . , f2s ∈Z[x1, . . . , xn] of degree ≤2d and of\ncoefficients bounded by 22M in absolute value, then:"},{"page":9,"text":"VPSPACE and a transfer theorem over the complex field\n9\n1. the variety V is defined by n + 1 polynomials g1, . . . , gn+1 ∈Z[x1, . . . , xn] of\ndegree ≤2d and of coefficients bounded by 22M+p(n,s,d) in absolute value;\n2. moreover, the coefficients of the gi are bitwise computable from those of the\nfj in working space Mp(n, s, d).\nProof. This is done by induction on s. Take p(n, s, d) = sq(n, d) where q(n, d) is\nthe polynomial defined in Lemma 3. The base case 2s ≤2(n + 1) follows from\nLemma 3. Suppose therefore that 2s > 2(n + 1). Call V1 and V2 the varieties\ndefined respectively by f1, . . . , f2s−1 and by f2s−1+1, . . . , f2s. Then V = V1 ∩V2\nand by induction hypothesis, V1 and V2 are both defined by n+1 polynomials of\ndegree ≤2d whose coefficients are bounded by 22M+(s−1)q(n,d) in absolute value\nand computable in space M(s −1)q(n, d).\nTherefore by Lemma 3, V is defined by n + 1 polynomials of degree ≤2d\nwhose coefficients are bounded by 22M+sq(n,d) in absolute value and computable\nin space Msq(n, d) as claimed in the proposition.\n⊓⊔\n4.2\nUnion of Varieties\nIn our case, however, the tests made by the algorithm of Section 5 are not exactly\nof the form studied in the previous section: instead of a single variety given by\ns polynomials, we have to decide “x ∈W?” when W ⊆Cn is the union of k\nvarieties. Of course, since the union is finite W is also a variety, but the encoding\nis not the same as above: now, k sets of s polynomials are given.\nA first naive approach is to define W = ∪iVi by the different products of the\npolynomials defining the Vi, but it turns out that there are too many products to\nbe dealt with. Instead, we will adopt a divide-and-conquer scheme as previously.\nLemma 4. There exists a polynomial q(n, d) such that, if V1 and V2 are two\nvarieties of Cn, each defined by n + 1 polynomials in Z[x1, . . . , xn], respectively\nf1, . . . , fn+1 and g1, . . . , gn+1, of degree ≤2d and of coefficients bounded by 22M\nin absolute value, then:\n1. the variety V = V1 ∪V2 is defined by n + 1 polynomials h1, . . . , hn+1 in\nZ[x1, . . . , xn] of degree ≤2d+1 and of coefficients bounded by 22M+q(n,d) in\nabsolute value;\n2. the coefficients of the hi are bitwise computable from those of the fj and gk\nin space Mq(n, d).\nProof. The variety V is defined by the (n + 1)2 polynomials figj for 1 ≤i, j ≤\nn + 1: these polynomials have degree ≤2d+1. Note moreover that there are at\nmost 2n(d+1) monomials of fixed degree δ ≤2d+1, therefore the coefficients of\nthe figj are a sum of at most 2n(d+1) products of integers of encoding size 2M.\nThus they are computable in space O(Mnd) from those of the fi and gj. This\nalso shows that the coefficients of the products figj are bounded in absolute\nvalue by 2n(d+1)22M+1 ≤22M+1+n(d+1). Applying Proposition 4 now enables to\nconclude if we take q(n, d) = 1 + n(d + 1) + p(n, log((n + 1)2), d + 1), where p is\nthe polynomial defined in Proposition 4.\n⊓⊔"},{"page":10,"text":"10\nPascal Koiran and Sylvain Perifel\nThe next proposition now follows by induction.\nProposition 5. There exists a polynomial r(n, s, k, d) such that, if V1, . . . , V2k ⊆\nCn are 2k varieties, Vi being defined by 2s polynomials f (i)\n1 , . . . , f (i)\n2s ∈Z[x1, . . . , xn]\nof degree ≤2d and of coefficients bounded by 22M in absolute value, then:\n1. the variety V\n= ∪2k\ni=1Vi is defined by n + 1 polynomials g1, . . . , gn+1 in\nZ[x1, . . . , xn] of degree ≤2d+k and whose coefficients are bounded in ab-\nsolute value by 22M+r(n,s,k,d);\n2. moreover, the coefficients of the gi are bitwise computable from those of the\nf (j)\nj′\nin space Mr(n, s, k, d).\nProof. We proceed by induction on k. Define r(n, s, k, d) = (k+1)(p(n, s, d+k)+\nq(n, d+k)), where p and q are defined in Proposition 4 and Lemma 4 respectively.\nThe base case k = 0 is merely an application of Proposition 4. For k > 0, we first\napply Proposition 4 to the Vi, so that each variety Vi is now defined by n + 1\npolynomials of degree ≤2d and whose coefficients are bounded in absolute value\nby 22M+p(n,s,d) and computable in space Mp(n, s, d). Let us group the varieties\nVi by pairs: call Wi = V2i−1 ∪V2i for 1 ≤i ≤2k−1. There are 2k−1 varieties Wi\nand we have V = ∪iWi. By Lemma 4, each variety Wi is defined by n + 1 poly-\nnomials of degree ≤2d+1, of coefficients of bitsize 2M+p(n,s,d)+q(n,d) and bitwise\ncomputable in space M(p(n, s, d) + q(n, d)). By induction hypothesis at rank\nk −1, V is defined by n+1 polynomials of degree ≤2d+1+(k−1), of coefficients of\nbitsize 2M+p(n,s,d)+q(n,d)+k(p(n,⌈log(n+1)⌉,d+k−1)+q(n,d+k−1)) ≤2M+r(n,s,k,d) and\nbitwise computable in space Mr(n, s, k, d). This proves the proposition.\n⊓⊔\nHere is the main consequence on membership tests to a union of varieties.\nCorollary 1. Let p(n) and q(n) be two polynomials. Suppose (fn( ̄x, ̄y, ̄z)) is a\nUniform VPSPACE0 family with | ̄x| = n, | ̄y| = p(n) and | ̄z| = q(n). For an integer\n0 ≤i < 2p(n), call V (n)\ni\n⊆Cn the variety defined by the polynomials fn( ̄x, i, j)\nfor 0 ≤j < 2q(n) (in this notation, i and j are encoded in binary).\nThen there exists a Uniform VPSPACE0 family gn( ̄x, ̄y, ̄z), where | ̄x| = n,\n| ̄y| = p(n) and | ̄z| = ⌈log(n + 1)⌉, such that\n∀ ̄x ∈Cn,\n∀k < 2p(n),\n \n ̄x ∈\nk[\ni=0\nV (n)\ni\n⇐⇒\nn\n^\nj=0\ngn( ̄x, k, j) = 0\n \n .\nProof. If (fn) is a Uniform VPSPACE0 family, by definition there exists a poly-\nnomial p(n) such that the degree of fn is bounded by 2p(n) and the absolute\nvalue of the coefficients by 22p(n). Therefore d, M, s and k are polynomially\nbounded in Proposition 5 and the space needed to compute the coefficients of\ngn is polynomial.\n⊓⊔"},{"page":11,"text":"VPSPACE and a transfer theorem over the complex field\n11\n5\nProof of the Main Theorem\nSign conditions are the main ingredient of the proof. Over C, we define the\n“sign” of a ∈C by 0 if a = 0 and 1 otherwise. Let us fix a family of polynomials\nf1, . . . , fs ∈Z[x1, . . . , xn]. A sign condition is an element S ∈{0, 1}s. Hence\nthere are 2s sign conditions. Intuitively, the i-th component of a sign condition\ndetermines the sign of the polynomial fi.\n5.1\nSatisfiable Sign Conditions\nThe sign condition of a point ̄x ∈Cn is the tuple S ̄x ∈{0, 1}s defined by\nS ̄x\ni = 0 ⇐⇒fi( ̄x) = 0. We say that a sign condition is satisfiable if it is the\nsign condition of some ̄x ∈Cn. As 0-1 tuples, sign conditions can be viewed as\nsubsets of {1, . . ., s}. Using a fast parallel sorting algorithm (e.g. Cole’s, [7]),\nwe can sort satisfiable sign conditions in polylogarithmic parallel time in a way\ncompatible with set inclusion (e.g. the lexicographic order). We now fix such\na compatible linear order on sign conditions and consider our satisfiable sign\nconditions S(1) < S(2) < . . . < S(N) sorted accordingly.\nThe key point resides in the following theorem, coming from the algorithm\nof [9]: there is a “small” number of satisfiable sign conditions and enumerating\nthem is “easy”.\nTheorem 3. Let f1, . . . , fs ∈Z[x1, . . . , xn] and d be their maximal degree. Then\nthe number of satisfiable sign conditions is N = (sd)O(n), and there is a uniform\nalgorithm working in space\n n log(sd)\n O(1) which, on boolean input f1, . . . , fs (in\ndense representation) and (i, j) in binary, returns the j-th component of the i-th\nsatisfiable sign condition.\nWhen log(sd) is polynomial in n, as will be the case, this yields a PSPACE\nalgorithm. If furthermore the coefficients of fi are computable in polynomial\nspace, we will then be able to use the satisfiable sign conditions in the coefficients\nof VPSPACE families, as in Lemma 5 below.\nLet us explain why we are interested in sign conditions. An arithmetic circuit\nperforms tests of the form f( ̄x) = 0 on input ̄x ∈Cn, where f is a polynomial.\nSuppose f1, . . . , fs is the list of all polynomials that can be tested in any possible\ncomputation. Then two elements of Cn with the same sign condition are simul-\ntaneously accepted or rejected by the circuit: the results of the tests are indeed\nalways the same for both elements.\nThus, instead of finding out whether ̄x ∈Cn is accepted by the circuit, it is\nenough to find out whether the sign condition of ̄x is accepted. The advantage\nresides in handling only boolean tuples (the sign conditions) instead of complex\nnumbers (the input ̄x). But we have to be able to find the sign condition of\nthe input ̄x. This requires first the enumeration of all the polynomials possibly\ntested in any computation of the circuit."},{"page":12,"text":"12\nPascal Koiran and Sylvain Perifel\n5.2\nEnumerating all Possibly Tested Polynomials\nIn the execution of an algebraic circuit, the values of some polynomials at the\ninput ̄x are tested to zero. In order to find the sign condition of the input ̄x, we\nhave to be able to enumerate in polynomial space all the polynomials that can\never be tested to zero in the computations of an algebraic circuit. This is done\nlevel by level as in [8, Th. 3] and [14].\nProposition 6. Let C be a constant-free algebraic circuit with n variables and\nof depth d.\n1. The number of different polynomials possibly tested to zero in the computa-\ntions of C is 2d2O(n).\n2. There exists an algorithm using work space (nd)O(1) which, on input C and\nintegers (i, j) in binary, outputs the j-th bit of the representation of the i-th\npolynomial.\nTogether with Theorem 3, this enables us to prove the following result which\nwill be useful in the proof of Proposition 7: in Uniform VPSPACE0 we can enu-\nmerate the polynomials as well as the satisfiable sign conditions.\nLemma 5. Let (Cn) be a uniform family of polynomial-depth algebraic circuits\nwith polynomially many inputs. Call d(n) the depth of Cn and i(n) the number\nof inputs. Let f (n)\n1\n, . . . , f (n)\ns\nbe all the polynomials possibly tested to zero by Cn\nas in Proposition 6, where s = 2O(nd(n)2). There are therefore N = 2O(n2d(n)2)\nsatisfiable sign conditions S(1), . . . , S(N) by Theorem 3.\nThen there exists a Uniform VPSPACE0 family (gn( ̄x, ̄y, ̄z)), where | ̄x| = i(n),\n| ̄y| = O(n2d(n)2) and | ̄z| = O(nd(n)2), such that for all 1 ≤i ≤N and 1 ≤j ≤\ns, we have:\ngn( ̄x, i, j) =\n(\n0\nif S(i)\nj\n= 1\nf (n)\nj\n( ̄x) otherwise.\n5.3\nFinding the Sign Condition of the Input\nIn order to find the sign condition S ̄x of the input ̄x ∈Cn, we will give a\npolynomial-time algorithm which tests some VPSPACE family for zero. Here is\nthe formalized notion of a polynomial-time algorithm with VPSPACE tests.\nDefinition 3. A polynomial-time algorithm with Uniform VPSPACE0 tests is\na Uniform VPSPACE0 family (fn(x1, . . . , xu(n))) together with a uniform fam-\nily (Cn) of constant-free polynomial-size algebraic circuits endowed with spe-\ncial test gates of indegree u(n), whose value is 1 on input (a1, . . . , au(n)) if\nfn(a1, . . . , au(n)) = 0 and 0 otherwise.\nObserve that a constant number of Uniform VPSPACE0 families can be used in\nthe preceding definition instead of only one: it is enough to combine them all in\none by using “selection variables”."},{"page":13,"text":"VPSPACE and a transfer theorem over the complex field\n13\nThe precise result we show now is the following. By the “rank” of a satisfiable\nsign condition, we merely mean its index in the fixed order on satisfiable sign\nconditions.\nProposition 7. Let (Cn) be a uniform family of algebraic circuits of polynomial\ndepth and with a polynomial number i(n) of inputs. There exists a polynomial-\ntime algorithm with Uniform VPSPACE0 tests which, on input ̄x ∈Ci(n), returns\nthe rank i of the sign condition S(i) of ̄x with respect to the polynomials g1, . . . , gs\ntested to zero by Cn given by Proposition 6.\nProof. Take the Uniform VPSPACE0 family (gn( ̄x, ̄y, ̄z)) as in Lemma 5: in essence,\ngn enumerates all the polynomials f1, . . . , fs possibly tested to zero in Cn and\nenumerates the N satisfiable sign conditions S(1) < . . . < S(N). The idea now is\nto perform a binary search in order to find the rank i of the sign condition of\nthe input ̄x.\nLet S(j) ∈{0, 1}s be a satisfiable sign condition. We say that S(j) is a candi-\ndate whenever ∀m ≤s, S(j)\nm = 0 ⇒fm( ̄x) = 0. Remark that the sign condition\nof ̄x is the smallest candidate. Call Vj the variety defined by the polynomi-\nals {fm|S(j)\nm\n= 0}: by definition of gn, Vj is also defined by the polynomials\ngn( ̄x, j, k) for k = 1 to s. Note that S(j) is a candidate if and only if ̄x ∈Vj.\nCorollary 1 combined with Lemma 5 asserts that tests of the form ̄x ∈∪k≤jVk\nare in Uniform VPSPACE0. They are used to perform a binary search by making\nj vary. In a number of steps logarithmic in N (i.e. polynomial in n), we find the\nrank i of the sign condition of ̄x.\n⊓⊔\n5.4\nA Polynomial-time Algorithm for PARC Problems\nLemma 6. Let (Cn) be a uniform family of constant-free polynomial-depth al-\ngebraic circuits. There is a (boolean) algorithm using work space polynomial in\nn which, on input i, decides whether the elements of the i-th satisfiable sign\ncondition S(i) are accepted by the circuit Cn.\nProof. We follow the circuit Cn level by level. For test gates, we compute the\npolynomial f to be tested. Then we enumerate the polynomials f1, . . . , fs as\nin Proposition 6 for the circuit Cn and we find the index j of f in this list.\nBy consulting the j-th bit of the i-th satisfiable sign condition with respect to\nf1, . . . , fs (which is done by the polynomial-space algorithm of Theorem 3), we\ntherefore know the result of the test and can go on like this until the output\ngate.\n⊓⊔\nTheorem 4. Let A ∈PAR0\nC. There exists a polynomial-time algorithm with\nUniform VPSPACE0 tests that decides A.\nProof. A is decided by a uniform family (Cn) of constant-free polynomial-depth\nalgebraic circuits. On input ̄x, thanks to Proposition 7 we first find the rank i of\nthe sign condition of ̄x with respect to the polynomials f1, . . . , fs of Proposition 6.\nThen we conclude by a last Uniform VPSPACE0 test simulating the polynomial-\nspace algorithm of Lemma 6 on input i.\n⊓⊔"},{"page":14,"text":"14\nPascal Koiran and Sylvain Perifel\nTheorem 2 follows immediately from this result. One could obtain other ver-\nsions of these two results by changing the uniformity conditions or the role of\nconstants.\nReferences\n1. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation.\nSpringer-Verlag, 1998.\n2. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over\nthe real numbers: NP-completeness, recursive functions and universal machines.\nBulletin of the American Mathematical Society, 21(1):1–46, 1989.\n3. P. B ̈urgisser. Completeness and Reduction in Algebraic Complexity Theory, vol-\nume 7 of Algorithms and Computation in Mathematics. Springer, 2000.\n4. P. B ̈urgisser. On implications between P-NP-hypotheses: Decision versus compu-\ntation in algebraic complexity. In Mathematical Foundations of Computer Science,\nvolume 2316 of Lecture Notes in Computer Science, pages 3–17. Springer, 2001.\n5. O. Chapuis and P. Koiran. Saturation and stability in the theory of computation\nover the reals. Annals of Pure and Applied Logic, 99:1–49, 1999.\n6. P. Charbit, E. Jeandel, P. Koiran, S. Perifel, and S. Thomass ́e. Finding a vector\northogonal to roughly half a collection of vectors. Available from http://perso.ens-\nlyon.fr/pascal.koiran/publications.html. Accepted for publication in Journal of\nComplexity, 2006.\n7. R. Cole. Parallel merge sort. SIAM J. Comput., 17(4):770–785, 1988.\n8. F. Cucker and D. Grigoriev. On the power of real Turing machines over binary\ninputs. SIAM Journal on Computing, 26(1):243–254, 1997.\n9. N. Fitchas, A. Galligo, and J. Morgenstern. Precise sequential and parallel com-\nplexity bounds for quantifier elimination over algebraically closed fields. Journal\nof Pure and Applied Algebra, 67:1–14, 1990.\n10. D. Grigoriev. Topological complexity of the range searching. Journal of Complexity,\n16:50–53, 2000.\n11. P. Koiran. Randomized and deterministic algorithms for the dimension of algebraic\nvarieties. In Proc. 38th IEEE Symposium on Foundations of Computer Science,\npages 36–45, 1997.\n12. P. Koiran. Circuits versus trees in algebraic complexity. In Proc. STACS 2000,\nvolume 1770 of Lecture Notes in Computer Science, pages 35–52. Springer, 2000.\n13. P. Koiran and S. Perifel.\nValiant’s model: from exponential sums to exponen-\ntial products. In Mathematical Foundations of Computer Science, volume 4162 of\nLecture Notes in Computer Science, pages 596–607. Springer-Verlag, 2006.\n14. P. Koiran and S. Perifel. VPSPACE and a transfer theorem over the reals. In\nProc. 24th Symposium on Theoretical Aspects of Computer Science, volume 4393\nof Lecture Notes in Computer Science, pages 417–428, 2007. Long version available\nfrom http://perso.ens-lyon.fr/pascal.koiran/publications.html.\n15. G. Malod. Polynˆomes et coefficients. PhD thesis, Universit ́e Claude Bernard Lyon\n1, July 2003. Available from http://tel.archives-ouvertes.fr/tel-00087399.\n16. G. Malod and N. Portier. Characterizing Valiant’s algebraic complexity classes.\nIn Mathematical Foundations of Computer Science, volume 4162 of Lecture Notes\nin Computer Science, pages 704–716. Springer-Verlag, 2006.\n17. B. Poizat. Les petits cailloux. Al ́eas, 1995.\n18. L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on\nTheory of Computing, pages 249–261, 1979."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"tures than {0, 1}. In this paper we will focus on the complex field (C, +, −, ×, =).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"recognize languages over C, that is, subsets of C∞= S","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"indegree 1, called test gates, are labelled “= 0?”. The size of a circuit C, in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"by a test gate is 0 if the value of its entry is ̸= 0 and 1 otherwise. Since we are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"Cn( ̄x, ̄a) = 1. The P-uniformity condition means that Cn can be built in time","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"Theorem 1. Let φ be a first-order formula over (C, 0, 1, +, −, ×, =) of the form","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"less than D(n, s, d) = 2p(n,s,d), and their coefficients are integers strictly bounded","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"is encoded in binary by α =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"fn( ̄x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"Proposition 2. Assuming the generalized Riemann hypothesis (GRH), VPnb =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"VPSPACE if and only if [P/poly = PSPACE/poly and VP = VNP].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"Proposition 3. Uniform VPSPACE0 = Uniform VP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"nb =⇒PSPACE = P-uniform NC.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"Remark 1. To the authors’ knowledge, the separation “PSPACE ̸= P-uniform NC”","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"Theorem 2 (main theorem). If Uniform VPSPACE0 = Uniform VP0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"C = P0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"C implies PARC =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"of Cn they define. Then for all coefficients (αi,j)i=1..s,j=1..n+1 ∈Cs(n+1) alge-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"braically independent over Q, the n + 1 linear combinations gj = Ps","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"i=1 αi,jfi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"structure (C, 0, 1, +, −, ×, =), containing only polynomials of degree less than","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"Assume furthermore that φ( ̄α) holds for all coefficients ̄α = (α1, . . . , αr) ∈Cr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"αi,jfj(x) = 0 ↔","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"fj(x) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"have their degree less than D = D(n, log(3(n + 1)), d + 1) and their coefficients","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"strictly bounded in absolute value by C = 22MD, where D(n, log(3(n + 1)), d +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"1) = 2p(n,log(3(n+1)),d+1) is defined in Theorem 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"r = (1 + 2(n + 1)2)p(n, log(3(n + 1)), d + 1), so that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"βi = 22M+2ir for 1 ≤i ≤2(n + 1)2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"nomial q(n, d) = 1 + 4(n + 1)2r (up to a multiplicative constant for the space","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"gi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"Proof. This is done by induction on s. Take p(n, s, d) = sq(n, d) where q(n, d) is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"defined respectively by f1, . . . , f2s−1 and by f2s−1+1, . . . , f2s. Then V = V1 ∩V2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"A first naive approach is to define W = ∪iVi by the different products of the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"1. the variety V = V1 ∪V2 is defined by n + 1 polynomials h1, . . . , hn+1 in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"conclude if we take q(n, d) = 1 + n(d + 1) + p(n, log((n + 1)2), d + 1), where p is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"= ∪2k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"i=1Vi is defined by n + 1 polynomials g1, . . . , gn+1 in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"Proof. We proceed by induction on k. Define r(n, s, k, d) = (k+1)(p(n, s, d+k)+","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"The base case k = 0 is merely an application of Proposition 4. For k > 0, we first","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"Vi by pairs: call Wi = V2i−1 ∪V2i for 1 ≤i ≤2k−1. There are 2k−1 varieties Wi","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"and we have V = ∪iWi. By Lemma 4, each variety Wi is defined by n + 1 poly-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"Uniform VPSPACE0 family with | ̄x| = n, | ̄y| = p(n) and | ̄z| = q(n). For an integer","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"Then there exists a Uniform VPSPACE0 family gn( ̄x, ̄y, ̄z), where | ̄x| = n,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"| ̄y| = p(n) and | ̄z| = ⌈log(n + 1)⌉, such that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"i=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"j=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"gn( ̄x, k, j) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"“sign” of a ∈C by 0 if a = 0 and 1 otherwise. Let us fix a family of polynomials","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"i = 0 ⇐⇒fi( ̄x) = 0. We say that a sign condition is satisfiable if it is the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"the number of satisfiable sign conditions is N = (sd)O(n), and there is a uniform","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"performs tests of the form f( ̄x) = 0 on input ̄x ∈Cn, where f is a polynomial.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"as in Proposition 6, where s = 2O(nd(n)2). There are therefore N = 2O(n2d(n)2)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"Then there exists a Uniform VPSPACE0 family (gn( ̄x, ̄y, ̄z)), where | ̄x| = i(n),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"| ̄y| = O(n2d(n)2) and | ̄z| = O(nd(n)2), such that for all 1 ≤i ≤N and 1 ≤j ≤","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"gn( ̄x, i, j) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"fn(a1, . . . , au(n)) = 0 and 0 otherwise.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"m = 0 ⇒fm( ̄x) = 0. Remark that the sign condition","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"= 0}: by definition of gn, Vj is also defined by the polynomials","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"gn( ̄x, j, k) for k = 1 to s. Note that S(j) is a candidate if and only if ̄x ∈Vj.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":38926,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}