Taylor Positivity of Ehrhart Polynomials

Abstract: 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 $\mathsf{A}<em>j(P;k)$ in the shifted expansion $$L_P(t)=\sum</em>{j=0}<sup>{d}\mathsf{A}_j(P;k)(t-k)<sup>j$$ 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 $h<sup>*$-vector, elementary symmetric functions, and Stirling numbers. (ii) We denote by $τ(P)$ and $τ<sup>+(P)$ the smallest nonnegative integral centers at which all Taylor coefficients are nonnegative and positive, respectively. If $s$ is the degree of the $h<sup>*$-polynomial, then $$0\leqτ(P)\leqτ<sup>+(P)\leq\min{\max{0,s-1},\lfloor\frac{d-1}{2}\rfloor}$$. 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 $λ<em>{\mathbb{R}}(f)$ denotes the largest real zero of $f(t)$, with value $-\infty$ when no such zero exists, then $ρ(P)=\max{0,\max</em>{0\leq j&lt;d}λ_{\mathbb{R}}!(L_P<sup>{(j)})}$. (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.