Unimodality Problems in Ehrhart Theory

TL;DR

This survey reviews the current understanding of unimodality in Ehrhart $h^*$-vectors, exploring their mathematical significance and highlighting open problems in the field.

Contribution

It compiles and discusses known results and open questions regarding the unimodality of Ehrhart $h^*$-vectors in lattice polytope theory.

Findings

Unimodality of Ehrhart $h^*$-vectors is a central open problem.

Connections exist between Ehrhart theory and algebraic combinatorics.

Many cases of unimodality are still unresolved.

Abstract

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.