Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension $34$.
@article{10_37236_7780,
author = {G\'abor Heged\"us and Akihiro Higashitani and Alexander Kasprzyk},
title = {Ehrhart polynomial roots of reflexive polytopes},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {1},
doi = {10.37236/7780},
zbl = {1497.52019},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7780/}
}
TY - JOUR
AU - Gábor Hegedüs
AU - Akihiro Higashitani
AU - Alexander Kasprzyk
TI - Ehrhart polynomial roots of reflexive polytopes
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/7780/
DO - 10.37236/7780
ID - 10_37236_7780
ER -
%0 Journal Article
%A Gábor Hegedüs
%A Akihiro Higashitani
%A Alexander Kasprzyk
%T Ehrhart polynomial roots of reflexive polytopes
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/7780/
%R 10.37236/7780
%F 10_37236_7780
Gábor Hegedüs; Akihiro Higashitani; Alexander Kasprzyk. Ehrhart polynomial roots of reflexive polytopes. The electronic journal of combinatorics, Tome 26 (2019) no. 1. doi: 10.37236/7780
