分裂拟阵的 $g$-多项式非负性
Nonnegativity of the $g$-polynomial of split matroids
- School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）
- School of Mathematical Sciences, Tianjin University of Technology（天津工业大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明分裂拟阵的 $g$-多项式系数非负，通过构造辅助分裂拟阵建立删除-收缩恒等式，从而在包含铺砌和余铺砌拟阵的类中证实了 Speyer 猜想。
AI中文摘要：
我们证明了每个分裂拟阵的 $g$-多项式具有非负系数，从而在某个对取子式封闭且包含所有铺砌拟阵和余铺砌拟阵的类中确立了 Speyer 猜想。我们的证明使用了通过构造一个辅助分裂拟阵得到的删除-收缩恒等式。我们首先证明每个简单、余简单、连通的分裂拟阵 $M$ 都有一个元素 $e$，使得删除 $M\setminus e$ 和收缩 $M/e$ 都是连通的。更一般地，设 $M$ 是秩为 $k$、基集为 $E$ 且 $|E|\ge4$ 的任意连通分裂拟阵。假设 $e\in E$ 使得 $M\setminus e$ 和 $M/e$ 都是连通的。对于每个 $v\in E\setminus\{e\}$，我们在 $E\setminus\{e,v\}$ 上构造一个秩为 $k-1$ 的连通初等分裂拟阵 $N_{e,v}$，满足 \\[ g_M(t)=g_{M\setminus e}(t)+g_{M/e}(t)+t\\,g_{N_{e,v}}(t). \\] 利用 $E$ 上的固定全序，我们规定了 $N_{e,v}$ 的适当循环闭包及其秩。该恒等式由 Ferroni 和 Schröter 的余赋值公式以及其修正多项式的递推关系得出，后者源自 Ferroni 对 Schubert 拟阵的容许 Delannoy 路径的枚举。由于右侧的三个拟阵具有更少的元素，该恒等式为非负性证明提供了归纳步骤。
英文摘要：
We prove that the $g$-polynomial of every split matroid has nonnegative coefficients, establishing Speyer's conjecture for a class closed under taking minors and containing all paving and copaving matroids. Our proof uses a deletion--contraction identity obtained by constructing an auxiliary split matroid. We first show that every simple, cosimple, connected split matroid $M$ has an element $e$ for which both the deletion $M\setminus e$ and the contraction $M/e$ are connected. More generally, let $M$ be any connected split matroid of rank $k$ on a ground set $E$ with $|E|\ge4$. Suppose that $e\in E$ is such that both $M\setminus e$ and $M/e$ are connected. For every $v\in E\setminus\{e\}$, we construct a connected elementary split matroid $N_{e,v}$ of rank $k-1$ on $E\setminus\{e,v\}$ satisfying \[ g_M(t)=g_{M\setminus e}(t)+g_{M/e}(t)+t\,g_{N_{e,v}}(t). \] Using a fixed total order on $E$, we prescribe the proper cyclic flats of $N_{e,v}$ and their ranks. The identity follows from the covaluative formula of Ferroni and Schröter together with recurrences for its correction polynomials, derived from Ferroni's enumeration of admissible Delannoy paths for Schubert matroids. Since all three matroids on the right have fewer elements, the identity supplies the induction step in the proof of nonnegativity.
