A genetic algorithm to search the space of Ehrhart $h^*$-vectors

TL;DR

This paper introduces a genetic algorithm designed to explore the space of Ehrhart $h^*$-vectors, successfully finding a counterexample to a conjecture about unimodality in high-dimensional lattice polytopes.

Contribution

It presents a novel genetic algorithm approach for searching $h^*$-vectors and provides the first known counterexample to a unimodality conjecture.

Findings

Discovered a 52-dimensional lattice polytope with a non-unimodal $h^*$-vector

The $h^*$-vector is a Cartesian product of two unimodal vectors

Answers negatively to a question by Ferroni and Higashitani

Abstract

We describe a genetic algorithm to find candidates for $h^*$-vectors satisfying given properties in the space of integers vectors of finite length. We use an implementation of such algorithm to find a 52-dimensional lattice polytope having a non-unimodal $h^*$-vector which is the Cartesian product of two lattice polytopes having unimodal $h^*$-vectors. This counterexample answers negatively to a question by Ferroni and Higashitani.