research/AMR-005-0015.md

metadata

id: AMR-005-0015
classification: SOLVED-IN-LITERATURE
wording_corrected: 'no'

AMR-005-0015 — Polynomial relations among triangle areas in a dissection of a square

Problem (corrected statement if needed)

Source: Serge Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1), 59–67 (2015), DOI 10.1007/s40598-014-0001-3, §13 "Areas and Dissections into Triangles", Problem 7 (verified against the publisher HTML at amj.math.stonybrook.edu and Crossref).

Original wording: Consider a partition of a square (of unspecified size) into $n$ triangles. Allowing small perturbations (each interior vertex has 2 degrees of freedom, each vertex on a side has 1, plus one for scaling), the moduli space $M$ of partitions with fixed combinatorics has $\dim M = n-1$. The map $M^{n-1}\to\mathbb{R}^n$ sending a partition to the ordered tuple of triangle areas is component-wise quadratic, so there is a polynomial relation among the areas $a_1,\dots,a_n$ depending only on the combinatorics. Problem 7: What can be said about this polynomial relation? For example, how to find the least degree of this polynomial in terms of the combinatorics of the partition?

The dataset transcription is a faithful paraphrase of the published problem; no correction needed. Motivation in the source: Monsky's theorem (a square cannot be dissected into an odd number of equal-area triangles; Stein–Szabo 1994, Monsky 1990).

Status / Literature

The problem is substantially answered by a research program of Aaron Abrams and James (Jamie) Pommersheim, plus related work. All items below were verified via Crossref API records and/or the arXiv API (DOIs/arXiv ids as stated).

1. A. Abrams, J. Pommersheim, "Spaces of Polygonal Triangulations and Monsky Polynomials", Discrete Comput. Geom. 51(1), 132–160 (2014). DOI 10.1007/s00454-013-9553-6; arXiv 2506.23444 (2025 arXiv upload of the published article). This paper — which Tabachnikov already cites — establishes: the areas of a (generalized) triangulation $T$ of a square satisfy a single irreducible homogeneous polynomial relation $p(T)$ depending only on the combinatorics of $T$, called the Monsky polynomial; and it gives an algorithm computing a lower bound on $\deg p(T)$, with several examples in which the algorithm computes the degree exactly. Since the relation ideal is principal, generated by the irreducible $p(T)$, the least-degree relation is exactly $p(T)$; hence "the least degree" = $\deg p(T)$.

2. A. Abrams, J. Pommersheim, "Generalized Dissections and Monsky's Theorem", Discrete Comput. Geom. 67(3), 947–983 (2022). DOI 10.1007/s00454-021-00354-9; arXiv 2006.04286. Establishes: the deformation space of generalized dissections (allowing flipped-orientation triangles) is an irreducible algebraic variety; Monsky's original relation polynomial $f$ can be chosen deformation-invariant, and a canonical pair of choices for $f$ is identified; and the striking structural theorem $$p(T)\equiv (a_1+a_2+\cdots +a_n{)}^d(\mathrm{m}\mathrm{o}\mathrm{d}2)p(T)\; \backslash equiv\; (a\_1+a\_2+\backslash cdots+a\_n)^\{d\}\; \backslash pmod\; 2$$ for some $d$ — i.e. modulo 2 the area relation is a pure power of the total area. This recovers and re-contextualizes Monsky's equidissection theorem: equal areas $a_i = 1/n$ with $n$ odd force $p(T)=0$ while the mod-2 form and a 2-adic valuation argument give a contradiction. Thus the polynomial relation "knows" the parity obstruction that motivated Tabachnikov's question.

3. A. Abrams, J. Pommersheim, "An Illustrated Encyclopedia of Area Relations", European J. Math. 9(3), art. 49 (2023). DOI 10.1007/s40879-023-00622-3; arXiv 2105.00563. Establishes: for fixed $l$, the set $\mathcal{E}_l$ of integer polynomials arising as irreducible factors of specializations of $p_T$ obtained by zeroing out all but $l$ variables is finite; $\mathcal{E}_l$ is computed explicitly for $l\le 4$; and in any dissection of a square into $l$ triangles the areas satisfy some polynomial in $\mathcal{E}_l$. Method: the rational "area map" from the drawing space to area space, and restrictions on the closure of its image from approaches to the base locus.

4. A. Abrams, J. Pommersheim, "Integrality Relations for Polygonal Dissections", Pacific J. Math. 330(2), 199–206 (2024). DOI 10.2140/pjm.2024.330.199. Establishes: in a dissection of a parallelogram, the area of any one triangle is integral over the ring generated by the other areas, with integrality relations invariant under deformation; a corollary is that the area polynomials (Monsky polynomials) for parallelograms have all leading coefficients equal to $\pm 1$; an analogous trapezoid theorem gives a new proof of Monsky's equidissection theorem.

5. Related: J.-P. Labbé, G. Rote, G. M. Ziegler, "Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles", Exp. Math. 29(3), 253–275 (2020). DOI 10.1080/10586458.2018.1459961. Uses Monsky polynomials computationally (all combinatorial triangulations of small size) to derive quantitative discrepancy bounds: in an odd dissection the areas cannot all be nearly equal, with explicit area-difference bounds. Background: P. Monsky, "On dividing a square into triangles", Am. Math. Monthly 77(2), 161–164 (1970), DOI 10.2307/2317329 (seen as a deposited reference in the records above).

Work done

• Read worklist/AMR-005-0015.md; fetched the publisher HTML of Tabachnikov's article and confirmed the item is §13, Problem 7, and that the dataset transcription is faithful.
• Verified every citation against the Crossref REST API (api.crossref.org/works/<DOI>) or the arXiv API: items 1–5 above, plus the source article's own DOI. The arXiv record 2506.23444 explicitly notes it is the post-publication upload of the 2014 DCG paper.
• Mathematical reasoning contributed (elementary checks and synthesis, no computation):
  • Why a unique least-degree relation exists. The area map $\alpha: M^{n-1}\to\mathbb{A}^n$ has constructible image of dimension $\le n-1$; its Zariski closure is a hypersurface (Abrams–Pommersheim show the deformation/drawing space is irreducible, so the closure is an irreducible hypersurface, defined over $\mathbb{Q}$ since the map is). The relation ideal in $\mathbb{Q}[a_1,\dots,a_n]$ is therefore principal, generated by a unique (up to scalar) irreducible polynomial $p(T)$; the least degree of any relation equals $\deg p(T)$. This reduces Tabachnikov's question to: describe $p(T)$ and compute $\deg p(T)$ from the combinatorics — precisely the content of papers 1–4.
  • Homogeneity. Scaling the square by $\lambda$ scales every triangle area by $\lambda^2$, so the image is a cone and $p(T)$ is homogeneous (consistent with paper 1's statement). With paper 4, one may normalize $p(T)$ to have integer coefficients and leading coefficients $\pm1$.
  • Hand-checked small cases. (i) $n=2$, square cut by a diagonal: $p = a_1 - a_2$, degree 1. (ii) $n=4$, one interior vertex joined to the four corners: writing $a_i$ for the triangle on side $i$, each $a_i = \tfrac12 s,d_i$ with $d_i$ the distance to that side, and opposite distances sum to $s$; hence $p = a_1 + a_3 - a_2 - a_4$, again degree 1. These match the theory: linear relations occur precisely when areas are constrained by affine "side-distance" bookkeeping; genuinely nonlinear Monsky polynomials appear for richer combinatorics (the smallest examples are catalogued in paper 3, which computes $\mathcal E_l$ for $l\le 4$).

Result

The problem is solved in the literature to the extent the question is posed. The definitive statements:

• The relation is given by a single irreducible homogeneous polynomial $p(T)\in\mathbb{Z}[a_1,\dots,a_n]$ (up to scalar), depending only on the combinatorics of the dissection: the Monsky polynomial (paper 1). It is invariant under deformation of the dissection (papers 2, 4), can be normalized to be monic with leading coefficients $\pm1$ in the parallelogram case (paper 4), and satisfies $p(T)\equiv (a_1+\cdots+a_n)^d \pmod 2$ (paper 2) — which explains Monsky's odd/even equidissection theorem as a corollary of the shape of the relation.
• Least degree: since the relation ideal is principal, the least degree is $\deg p(T)$. Paper 1 gives a combinatorial algorithm that computes a lower bound on $\deg p(T)$ and computes the exact degree in worked examples; paper 3 adds finiteness and explicit computation of all low-width relations ($\mathcal E_l$ for $l\le 4$), and paper 5 shows the degrees/coefficients are effectively computable for all triangulations of modest size by direct enumeration. The general qualitative answer to "what can be said" is thus complete; the specific degree question has an algorithmic (not closed-form) answer.

What remains

• No known closed combinatorial formula for $\deg p(T)$ (or for $p(T)$ itself) valid for all combinatorial dissections; the 2014 algorithm yields a lower bound, proved exact in examples, but I did not find a published theorem that it is always exact. Producing such a formula — or proving the lower bound always equals the degree — is the natural next step.
• $\mathcal E_l$ is computed only for $l\le 4$; extending the encyclopedia, and understanding growth/complexity of $p(T)$ as $n\to\infty$, is open.
• Analogues for dissections of other polygons are partly covered (parallelograms, trapezoids in paper 4; the 2014 paper works with $n$-gons), but a systematic theory for general polygonal regions and higher-dimensional (simplex-volume) analogues appears largely undeveloped.
• Caveat: my summary of each paper's content is based on its verified abstract and bibliographic record, not on a line-by-line reading of the full texts; the precise hypotheses (e.g. the class of "generalized triangulations" needed for irreducibility/uniqueness statements) should be checked in the papers themselves before being quoted in a proof.