串并联扩展与 Ehrhart 正性
Series-parallel extensions and Ehrhart positivity
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中文总结 AI 辅助
本文证明串并联拟阵及其多路径扩展的 Ehrhart 正性，推导系数公式与格点计数递归，并构造反例否证相关猜想。
中文摘要 AI 辅助
我们证明了串并联拟阵的 Ehrhart 正性，并将此结果推广到多路径拟阵的所有串并联扩展。推导出了串并联拟阵的 Ehrhart 系数公式，并利用该公式证明了严格正性。建立了一个基于加权格点计数的扩展准则。推导出了格点计数的递归公式，并证明该递归保持一类具有非负系数的多项式。最后，我们构造了一个具有负 Ehrhart 系数的连通横截拟阵，从而否证了 Ferroni、Morales 和 Panova 的猜想：每个其基多面体可细分为串并联拟阵基多面体的连通拟阵都是 Ehrhart 正的。
英文摘要
We prove Ehrhart positivity for series-parallel matroids and extend this result to all series-parallel extensions of multi-path matroids. A formula for the Ehrhart coefficients of series-parallel matroids is derived and used to prove strict positivity. An extension criterion based on weighted lattice-point counts is established. Recursive formulas for lattice-point counts are derived, and the recursion is shown to preserve a class of polynomials with nonnegative coefficients. We finally construct a connected transversal matroid with a negative Ehrhart coefficient, disproving the conjecture of Ferroni, Morales, and Panova that every connected matroid whose base polytope admits a subdivision into series-parallel matroid base polytopes is Ehrhart positive.
