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<p>A bimatroid is a matroid-like generalization of the collection of regular minors of a matrix. In this article, we use the theory of Lorentzian polynomials to study the logarithmic concavity of natural sequences associated with bimatroids. Bimatroids can be used to characterize morphisms of matroids and this observation (originally due to Kung) allows us to prove a weak version of logarithmic concavity of the number of bases of a morphism of matroids. This is weaker than the original result by Eur and Huh; it nevertheless provides us with a new perspective on Mason’s log-concavity conjecture for independent sets of matroids. We finally show that for realizable bimatroids, the regular minor polynomial is a volume polynomial. Applied to morphisms of matroids, this shows that the weak basis generating polynomial of a morphism is a volume polynomial; this confirms a conjecture of Eur–Huh for morphisms of nullity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\le 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and gives an algebro-geometric explanation for Mason’s log-concavity conjecture in the realizable case.</p>

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Logarithmic Concavity of Bimatroids

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  • 发表日期：

    2025年8月14日

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  • Felix Röhrle,
  • Martin Ulirsch

摘要

A bimatroid is a matroid-like generalization of the collection of regular minors of a matrix. In this article, we use the theory of Lorentzian polynomials to study the logarithmic concavity of natural sequences associated with bimatroids. Bimatroids can be used to characterize morphisms of matroids and this observation (originally due to Kung) allows us to prove a weak version of logarithmic concavity of the number of bases of a morphism of matroids. This is weaker than the original result by Eur and Huh; it nevertheless provides us with a new perspective on Mason’s log-concavity conjecture for independent sets of matroids. We finally show that for realizable bimatroids, the regular minor polynomial is a volume polynomial. Applied to morphisms of matroids, this shows that the weak basis generating polynomial of a morphism is a volume polynomial; this confirms a conjecture of Eur–Huh for morphisms of nullity \(\le 1\) ≤ 1 and gives an algebro-geometric explanation for Mason’s log-concavity conjecture in the realizable case.