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arXiv · 2608.21507

A very ample lattice polytope with a non-unimodal $h^*$-vector

Johannes Hofscheier·Vadym Kurylenko·Benjamin Nill
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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 $h^*$-vector is non-unimodal. Here, the $h^*$-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 $h^*$-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 Laso\'n and Michalek.

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Johannes Hofscheier, Vadym Kurylenko, Benjamin Nill. 2026-08-21. A very ample lattice polytope with a non-unimodal $h^*$-vector. https://arxiv.org/abs/2608.21507

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