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| id: AMR-005-0015 |
| classification: SOLVED-IN-LITERATURE |
| wording_corrected: no |
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| # AMR-005-0015 — Polynomial relations among triangle areas in a dissection of a square |
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| ## Problem (corrected statement if needed) |
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| Source: Serge Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1), 59–67 (2015), DOI [10.1007/s40598-014-0001-3](https://doi.org/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). |
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| 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?** |
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| 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). |
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| ## Status / Literature |
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| 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). |
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| 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](https://doi.org/10.1007/s00454-013-9553-6); arXiv [2506.23444](https://arxiv.org/abs/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" = $\deg p(T)$. |
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| 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](https://doi.org/10.1007/s00454-021-00354-9); arXiv [2006.04286](https://arxiv.org/abs/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 |
| $$p(T) \equiv (a_1+a_2+\cdots+a_n)^{d} \pmod 2$$ |
| 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. |
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| 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](https://doi.org/10.1007/s40879-023-00622-3); arXiv [2105.00563](https://arxiv.org/abs/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. |
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| 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](https://doi.org/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. |
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| 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](https://doi.org/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). |
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| ## Work done |
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| - 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 $l\le 4$). |
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| ## Result |
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| The problem is solved in the literature to the extent the question is posed. The definitive statements: |
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| - 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. |
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| ## What remains |
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| - 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. |
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