Taylor Positivity of Ehrhart Polynomials
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 $\tau(P)$ and $\tau^+(P)$ 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 $0\leq\tau(P)\leq\tau^+(P)\leq\min\{\max\{0,s-1\},\lfloor\frac{d-1}{2}\rfloor\}$. 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 $\rho(P)$ be the smallest nonnegative real center such that the Taylor coefficients are nonnegative. If $\lambda_{\mathbb{R}}(f)$ denotes the largest real zero of $f(t)$, with value $-\infty$ when no such zero exists, then $\rho(P)=\max\{0,\max_{0\leq j<d}\lambda_{\mathbb{R}}\!(L_P^{(j)})\}$. (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.