A very ample lattice polytope with a non-unimodal $h^{\ast }$-vector

Abstract

Lattice polytopes are called very ample if for every sufficiently large $k$ every lattice point of height $k$ in the cone over the lattice polytope is the sum of $k$ lattice points of height $1$. This is a weakening of the well-known integer decomposition property (also called IDP). We give an example of a very ample lattice polytope whose $h^{\ast }$-vector is non-unimodal. Here, the $h^{\ast }$-vector is the coefficient vector of the numerator of the Ehrhart series of the lattice polytope. This answers a question of Ferroni and Higashitani, as well as a related question by Balletti. The main question whether IDP lattice polytopes have unimodal $h^{\ast }$-vector is still open. The example was found using ChatGPT 5.6 Sol. It is just the Cartesian square of a lattice polytope belonging to a class of very ample examples constructed by Lasoń and Michalek.

4 pages