Taylor Positivity of Ehrhart Polynomials
Abstract
Let P be a d-dimensional lattice polytope with Ehrhart polynomial LP(t). Motivated by the study of Ehrhart positivity and magic positivity, we investigate the Taylor coefficients Aj(P;k) in the shifted expansion LP(t)=∑j=0dAj(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≤τ(P)≤τ+(P)≤min{max{0,s−1},⌊(d−1)/2⌋}. As an application, we slightly improve an upper bound due to Beck, De Loera, Develin, Pfeifle, and Stanley. That is, every real root of LP(t) lies in [−d,⌊(d−1)/2⌋). (iii) Let ρ(P) be the smallest nonnegative real center such that the Taylor coefficients are nonnegative. If λℝ(f) denotes the largest real zero of f(t), with value −∞ when no such zero exists, then ρ(P)=max{0,max0≤ j<dλℝ (LP(j))}. (iv) We establish structural properties of the Taylor coefficients Aj(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.
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BibTeX
@misc{liu2026taylor,
title = {{Taylor Positivity of Ehrhart Polynomials}},
author = {Feihu Liu and Zihao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.03327},
eprint = {2609.03327},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03327},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Taylor Positivity of Ehrhart Polynomials
AU - Feihu Liu
AU - Zihao Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.03327
UR - https://arxiv.org/abs/2609.03327
AB - 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 A_j(P;k) in the shifted expansion L_P(t)=∑_j=0^dA_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≤τ(P)≤τ^+(P)≤min{max{0,s−1},⌊(d−1)/2⌋}. 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,⌊(d−1)/2⌋). (iii) Let ρ(P) be the smallest nonnegative real center such that the Taylor coefficients are nonnegative. If λ_ℝ(f) denotes the largest real zero of f(t), with value −∞ when no such zero exists, then ρ(P)=max{0,max_0≤ j<dλ_ℝ (L_P^(j))}. (iv) We establish structural properties of the Taylor coefficients 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.
ER -