发表机构
Nankai University; Beijing Institute of Technology（南开大学; 北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究格多面体Ehrhart多项式的泰勒正性，推导泰勒系数的精确公式，确定相关中心的上界，建立系数结构性质，改进Ehrhart多项式实根的上界，为开放问题提供部分解答。
AI 中文摘要
设$P$是一个$d$维格多面体，其Ehrhart多项式为$L_P(t)$。受Ehrhart正性与魔术正性研究的启发，我们研究在实中心$k$处的移位展开$L_P(t)=\u2211_{j=0}^{d}\u1d41_j(P;k)(t-k)^j$中的泰勒系数$\u1d41_j(P;k)$。本文得到以下四个主要结果：(i) 我们根据普通Ehrhart系数、$h^*$向量、初等对称函数和斯特林数，给出了这些系数的精确公式；(ii) 我们用$\u03c4(P)$和$\u03c4^+(P)$分别表示使所有泰勒系数非负和为正的最小非负整数中心，若$s$是$h^*$多项式的次数，则有$0\u2264\u03c4(P)\u2264\u03c4^+(P)\u2264\min\{\max\{0,s-1\},\lfloor\frac{d-1}{2}\rfloor\}$；作为应用，我们略微改进了Beck、De Loera、Develin、Pfeifle和Stanley给出的上界，即$L_P(t)$的每个实根都位于$[-d,\lfloor\frac{d-1}{2}\rfloor)$内；(iii) 设$\u03c1(P)$是使泰勒系数非负的最小非负实中心，若$\u03bb_{\mathbb{R}}(f)$表示$f(t)$的最大实零点，无零点时取值为$-\u221e$，则$\u03c1(P)=\max\{0,\max_{0\u2264 j<d}\u03bb_{\mathbb{R}}(L_P^{(j)})\}$；(iv) 我们建立了泰勒系数$\u1d41_j(P;k)$的结构性质，包括导数交错、回文反射对称以及Laguerre和Newton不等式。最后，这些结果为美国数学研究所网站列出的一个开放问题提供了系统的部分解答。
英文摘要
Let $P$ be a $d$-dimensional lattice polytope with Ehrhart polynomial $L_P(t)$. Motivated by the study of Ehrhart positivity and magic positivity, we investigate the Taylor coefficients $\mathsf{A}_j(P;k)$ in the shifted expansion $L_P(t)=\sum_{j=0}^{d}\mathsf{A}_j(P;k)(t-k)^j$ about a real center $k$. In this paper, we obtain the following four main results. (i) We give exact formulas for these coefficients in terms of the ordinary Ehrhart coefficients, the $h^*$-vector, elementary symmetric functions, and Stirling numbers. (ii) We denote by $τ(P)$ and $τ^+(P)$ the smallest nonnegative integral centers at which all Taylor coefficients are nonnegative and positive, respectively. If $s$ is the degree of the $h^*$-polynomial, then $0\leqτ(P)\leqτ^+(P)\leq\min\{\max\{0,s-1\},\lfloor\frac{d-1}{2}\rfloor\}$. As an application, we slightly improve an upper bound due to Beck, De Loera, Develin, Pfeifle, and Stanley. That is, every real root of $L_P(t)$ lies in $[-d,\lfloor\frac{d-1}{2}\rfloor)$. (iii) Let $ρ(P)$ be the smallest nonnegative real center such that the Taylor coefficients are nonnegative. If $λ_{\mathbb{R}}(f)$ denotes the largest real zero of $f(t)$, with value $-\infty$ when no such zero exists, then $ρ(P)=\max\{0,\max_{0\leq j<d}λ_{\mathbb{R}}\!(L_P^{(j)})\}$. (iv) We establish structural properties of the Taylor coefficients $\mathsf{A}_j(P;k)$, including derivative interlacing, palindromic reflection symmetries, and Laguerre and Newton inequalities. As a final note, these results provide a systematic partial answer to an open problem listed on the website of the American Institute of Mathematics.
Comments42 pages
