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arxiv: 2401.03383 · v4 · pith:ROIVHAVQnew · submitted 2024-01-07 · 🧮 math.CO

On the Ehrhart Theory of Generalized Symmetric Edge Polytopes

Robert Davis , Akihiro Higashitani , Hidefumi Ohsugi This is my paper

Pith reviewed 2026-05-24 04:34 UTC · model grok-4.3

classification 🧮 math.CO MSC 52B2005B35
keywords symmetric edge polytoperegular matroidEhrhart polynomialgamma-vectorgamma-nonnegativitylattice polytopeGröbner basis
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The pith

Generalized symmetric edge polytopes of regular matroids need not have nonnegative gamma-vectors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that symmetric edge polytopes, when generalized from graphs to regular matroids, can produce Ehrhart h*-polynomials whose gamma-vectors contain negative entries. Explicit constructions are given that demonstrate this failure of gamma-nonnegativity while showing the constructed polytopes become gamma-nonnegative after exactly two elements are deleted from the matroid, recovering ordinary graph SEPs. Combinatorial arguments and Gröbner basis techniques are used to carry over other known SEP properties to the generalized setting. The examples supply supporting evidence for the conjecture that ordinary graph SEPs always have nonnegative gamma-vectors.

Core claim

Generalized symmetric edge polytopes associated to regular matroids are not necessarily gamma-nonnegative. Explicit matroid examples are constructed whose gamma-vectors contain negative entries. These polytopes are nearly gamma-nonnegative in the sense that deleting exactly two elements from the matroid yields SEPs for graphs that are gamma-nonnegative.

What carries the argument

The gamma-vector of the symmetric h*-polynomial of the generalized symmetric edge polytope of a regular matroid, which can take negative values in the constructed examples.

If this is right

  • Other known properties of ordinary SEPs extend to generalized SEPs via combinatorial and Gröbner basis methods.
  • The Ohsugi-Tsuchiya conjecture on gamma-nonnegativity holds for the ordinary graph case in the tested examples.
  • Deleting two elements from the matroid turns the constructed non-gamma-nonnegative generalized SEP into a gamma-nonnegative ordinary SEP.
  • The failure of gamma-nonnegativity occurs only in a narrow way for these regular matroids.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Gamma-nonnegativity appears to be a property specific to graphic matroids rather than all regular matroids.
  • The deletion technique may identify a larger class of matroids for which generalized SEPs satisfy gamma-nonnegativity.
  • Similar narrow failures could appear for other Ehrhart invariants when moving from graphs to regular matroids.

Load-bearing premise

The explicit matroid constructions are regular matroids and the gamma-vectors of their polytopes have been correctly computed to contain negative entries.

What would settle it

An independent computation of the gamma-vector for one of the paper's explicit matroid examples that shows all entries are nonnegative.

Figures

Figures reproduced from arXiv: 2401.03383 by Akihiro Higashitani, Hidefumi Ohsugi, Robert Davis.

Figure 1
Figure 1. Figure 1: The graphs Γ(3), left, and Γ(4), right. Remark 5.2. We can obtain Γ(n) as the graph representing M∗ (K3,n)−{e, e′} in an alternate way. Although K3,n/e is not planar for any edge e of K3,n as explained in the proof of Proposition 4.1 (ii), we can get a planar graph G˜ by the contraction of an edge e ′ of K3,n/e such that e and e ′ were incident to the same degree-3 vertex in K3,n. By taking the dual graph … view at source ↗
Figure 2
Figure 2. Figure 2: On the left, K3,4 with a choice of e and e ′ labeled such that Γ(4) is an underlying graph of M∗ (K3,4) − {e, e′}. On the right, a planar representation of K3,4/{e, e′} in solid black lines, with its dual, Γ(4), overlaid in dashed gray lines. In what follows, we concentrate on the graph Γ(n + 1). Let ee = u1un+1. To describe its spanning trees, it will be helpful to refer to a specific planar embedding of … view at source ↗
Figure 3
Figure 3. Figure 3: A spanning tree of Γ(5), indicated by the solid edges. The edges u1v1 and v1u2 form a chordless pair. The edges u2v2 and u2u3 form a chorded pair of type α, while the edges u4v4 and u4u5 form a chorded pair of type β. The edge v3u4 is an unpaired edge. u1 v1 u2 v2 u3 e v3 u4 u1 v1 u2 v2 u3 e v3 u4 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two spanning trees of Γ(4) that are members of the triangulating tree set, as described in Proposition 5.3. For a spanning tree T of Γ(n + 1), let c(T ) denote the number of chords uiui+1 of T and let ǫ(T ) = ( 1 if ee ∈ T 0 otherwise. Lemma 5.5. Let T be a member of our triangulating tree set. (i) If ǫ(T ) = 1, then T contains a total of n − 1 chordless pairs and chorded pairs of edges as well as exactly … view at source ↗
Figure 5
Figure 5. Figure 5: Examples of modified spanning trees in Γ(4). The original spanning trees and their corresponding modified trees are placed above and below each other. See [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

The symmetric edge polytope (SEP) of a (finite, undirected) graph is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. SEPs have been studied extensively in the past twenty years. Recently, T\'othm\'er\'esz and, independently, D'Al\'i, Juhnke-Kubitzke, and Koch generalized the definition of an SEP to regular matroids, which are the matroids that can be represented by totally unimodular matrices. Generalized SEPs are known to have symmetric Ehrhart $h^*$-polynomials, and Ohsugi and Tsuchiya conjectured that (ordinary) SEPs have nonnegative $\gamma$-vectors. In this article, we use combinatorial and Gr\"obner basis techniques to extend additional known properties of SEPs to generalized SEPs. Along the way, we show that generalized SEPs are not necessarily $\gamma$-nonnegative by providing explicit examples. We prove that the polytopes we construct are ``nearly'' $\gamma$-nonnegative in the sense that, by deleting exactly two elements from the matroid, one obtains SEPs for graphs that are $\gamma$-nonnegative. This provides further evidence that Ohsugi and Tsuchiya's conjecture holds in the ordinary case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper extends several known properties of symmetric edge polytopes (SEPs) of graphs to generalized SEPs of regular matroids, using combinatorial arguments and Gröbner basis techniques to study their Ehrhart h*-polynomials. It supplies explicit regular matroid examples showing that generalized SEPs need not be γ-nonnegative, while proving that deletion of exactly two elements yields ordinary graph SEPs whose γ-vectors are nonnegative, thereby furnishing supporting evidence for the Ohsugi–Tsuchiya conjecture in the graph case.

Significance. If the explicit matroid constructions and the associated γ-vector computations hold, the result is significant: it cleanly separates the generalized setting (where γ-nonnegativity fails) from the ordinary graph setting (where the conjecture may still hold) and isolates a precise “nearly γ-nonnegative” deletion property. The deployment of Gröbner bases to obtain the h*-polynomials and the explicit counterexamples constitute concrete, falsifiable contributions.

major comments (3)
  1. [matroid constructions (around the explicit examples)] The central negative result rests on the claim that the constructed matroids are regular and that their associated polytopes have negative γ-entries. The manuscript must therefore supply, for each example, either an explicit totally unimodular representation matrix or a self-contained proof of regularity; without this, the applicability of the generalized-SEP definition cannot be verified.
  2. [γ-vector calculations] The conversion from the computed h*-vector to the γ-vector must be exhibited in full for the counterexamples, including the precise negative coefficients and the binomial inversion steps. Any omission here directly affects the load-bearing claim that γ-nonnegativity fails.
  3. [deletion property] The deletion argument asserts that removing exactly two elements produces ordinary graph SEPs that are γ-nonnegative. The manuscript should identify the resulting graphs explicitly and confirm (by citation or direct computation) that their γ-vectors are indeed nonnegative; this step is essential to the “nearly γ-nonnegative” statement.
minor comments (2)
  1. Notation for the ground set and the deletion operation should be introduced once and used consistently throughout.
  2. A short table summarizing the h*- and γ-vectors of the counterexamples would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will incorporate the requested clarifications and explicit data into the revised manuscript.

read point-by-point responses

  1. Referee: [matroid constructions (around the explicit examples)] The central negative result rests on the claim that the constructed matroids are regular and that their associated polytopes have negative γ-entries. The manuscript must therefore supply, for each example, either an explicit totally unimodular representation matrix or a self-contained proof of regularity; without this, the applicability of the generalized-SEP definition cannot be verified.

    Authors: We agree that explicit verification strengthens the paper. In the revision we will append the totally unimodular representation matrices for each counterexample matroid together with a short check that every square submatrix has determinant in {−1,0,1}. revision: yes

  2. Referee: [γ-vector calculations] The conversion from the computed h*-vector to the γ-vector must be exhibited in full for the counterexamples, including the precise negative coefficients and the binomial inversion steps. Any omission here directly affects the load-bearing claim that γ-nonnegativity fails.

    Authors: We will expand the relevant section to display the complete binomial-inversion formulas, the intermediate h*-vectors, and the resulting γ-vectors with the explicit negative entries highlighted for each counterexample. revision: yes

  3. Referee: [deletion property] The deletion argument asserts that removing exactly two elements produces ordinary graph SEPs that are γ-nonnegative. The manuscript should identify the resulting graphs explicitly and confirm (by citation or direct computation) that their γ-vectors are indeed nonnegative; this step is essential to the “nearly γ-nonnegative” statement.

    Authors: In the revised manuscript we will name the two deleted elements for each example, state the resulting graphic matroids (i.e., the explicit graphs), and either cite the known γ-nonnegativity results for those graphs or include the short direct γ-vector computations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit constructions are self-contained

full rationale

The central claim (generalized SEPs need not be γ-nonnegative) rests on explicit regular matroid constructions whose γ-vectors are computed directly via combinatorial and Gröbner basis techniques applied to the matroid polytopes. These computations do not reduce by definition or self-citation to the target result; they are independent verifications against standard Ehrhart and matroid definitions. The additional claim that deleting two elements yields ordinary graph SEPs that are γ-nonnegative draws on known cases or direct verification outside the fitted values of the counterexamples themselves. No load-bearing step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions and known properties of regular matroids and Ehrhart theory without introducing free parameters or new entities.

axioms (2)
  • domain assumption Regular matroids admit representations by totally unimodular matrices
    Basis for the generalization stated in the abstract.
  • standard math Symmetric edge polytopes have symmetric Ehrhart h*-polynomials
    Known property extended to the generalized case.

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Forward citations

Cited by 1 Pith paper

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  1. Unimodular polytopes and column number bounds on polytopal totally unimodular matrices via Seymour's decomposition theorem

    math.CO 2024-05 unverdicted novelty 6.0

    A sharp upper bound is established on distinct columns of unit-sum polytopal totally unimodular matrices and on vertices of unimodular polytopes.

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