发表机构
Center for Combinatorics, LPMC,Nankai University; School of Mathematics and Statistics,Beijing Institute of Technology（南开大学组合数学中心; 北京理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了 Morris 常数项变体的生成函数 $h_n^*(y)$ 具有正整系数、实根性、Gamma 正性、回文性、单峰性和超对数凹性，从而证实了 Xin 和 Zhang 的猜想。
AI 中文摘要
Beck 和 Pixton 将 Birkhoff 多胞形的 Ehrhart 多项式表示为若干多元有理函数常数项的加权和。Xin 和 Zhang 研究了一类常数项 $h_n(t)$，它可被视为 Morris 常数项的变体。他们证明了 $h_n(t)$ 是次数为 $(n-1)^2$ 的多项式，并获得了涉及 Morris 常数项恒等式的许多优良性质。设 $h_n^*(y)=(1-y)^{(n-1)^2+1}\sum_{t\geq0}h_n(t)y^t$。对于固定的 $n\geq 3$，我们得到以下三个主要结果：(i) $h_n^*(y)$ 是系数为正整数的多项式。(ii) $h_n^*(y)$ 是实根的。特别地，它的所有根都是非正实数。(iii) $h_n^*(y)$ 是 Gamma 正的。此外，$h_n^*(y)$ 是回文的、单峰的以及超对数凹的。这证实了 Xin 和 Zhang 关于 $h_n^*(y)$ 的猜想。作为副产品，我们证明了与 $h_n^*(y)$ 相关联的 Gamma 多项式的每个根都是负实数。
英文摘要
Beck and Pixton expressed the Ehrhart polynomial of the Birkhoff polytope as a weighted sum of constant terms of several multivariate rational functions. Xin and Zhang studied a class of constant terms $h_n(t)$, which can be regarded as a variation of the Morris constant term. They proved that $h_n(t)$ is a polynomial of degree $(n-1)^2$ and obtained many nice properties involving the Morris constant term identity. Let $h_n^*(y)=(1-y)^{(n-1)^2+1}\sum_{t\geq0}h_n(t)y^t$. For fixed $n\geq 3$, we obtain the following results for $h_n^*(y)$: (i): $h_n^*(y)$ is a polynomial with positive integer coefficients. (ii): $h_n^*(y)$ is real-rooted. In particular, all its roots are non-positive real numbers. (iii): $h_n^*(y)$ is Gamma-positive. Furthermore, $h_n^*(y)$ is palindromic, unimodal, and ultra log-concave. This confirms Xin and Zhang's conjecture regarding $h_n^*(y)$. In order to resolve this conjecture, we also developed an operator on the space of symmetric polynomials that preserves real stability. As a byproduct, we prove that every root of a Gamma polynomial associated with $h_n^*(y)$ is a negative real number.
