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<p>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, Tóthmérész and, independently, D’Alí, 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(h^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-polynomials, and Ohsugi and Tsuchiya conjectured that (ordinary) SEPs have nonnegative <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-vectors. In this article, we use combinatorial and Gröbner basis techniques to extend additional known properties of SEPs to generalized SEPs. Along the way, we show that generalized SEPs are not necessarily <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-nonnegative by providing explicit examples. We prove that the polytopes we construct are “nearly” <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-nonnegative in the sense that, by deleting two particular elements from the matroid, one obtains SEPs for graphs that are <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-nonnegative. This provides further evidence that Ohsugi and Tsuchiya’s conjecture holds in the ordinary case.</p>

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On the Ehrhart Theory of Generalized Symmetric Edge Polytopes

  • 期刊论文

  • 发表日期：

    2026年6月23日

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  • Robert Davis,
  • Akihiro Higashitani,
  • Hidefumi Ohsugi

摘要

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, Tóthmérész and, independently, D’Alí, 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 \(h^*\) h ∗ -polynomials, and Ohsugi and Tsuchiya conjectured that (ordinary) SEPs have nonnegative \(\gamma \) γ -vectors. In this article, we use combinatorial and Gröbner basis techniques to extend additional known properties of SEPs to generalized SEPs. Along the way, we show that generalized SEPs are not necessarily \(\gamma \) γ -nonnegative by providing explicit examples. We prove that the polytopes we construct are “nearly” \(\gamma \) γ -nonnegative in the sense that, by deleting two particular elements from the matroid, one obtains SEPs for graphs that are \(\gamma \) γ -nonnegative. This provides further evidence that Ohsugi and Tsuchiya’s conjecture holds in the ordinary case.