Ehrhart h*-polynomials and Regular Unimodular Triangulations

Abstract

The Ehrhart series of a convex lattice polytope P is the generating function that records the number of lattice points in the k^th dilate of P for each nonnegative integer k.  A theorem of Stanley tells us that the rational form of the Ehrhart series has a numerator polynomial, called the h*-polynomial of P, that has only nonnegative integer coefficients.  The coefficients of the h*-polynomial often admit a combinatorial interpretation, and consequently, conditions on P that guarantee symmetry and/or unimodality of these coefficients have been studied extensively.  Provided that the h*-polynomial is symmetric, a famous theorem of Bruns and Römer says it will also be unimodal whenever P admits a regular unimodular triangulation.  In fact, many of the well-studied families of polytopes known to have symmetric and unimodal h*-polynomials have some regular unimodular triangulation.  In this talk, we will first describe techniques from toric algebra that can be used to identify regular unimodular triangulations of polytopes, and we will describe some important examples in the literature that arise in this fashion.  Following this, we will discuss the potential for proving unimodality of h*-polynomials by means that escape this powerful theorem.