Katharina Jochemko: h*-polynomials of zonotopes

Abstract

The Ehrhart polynomial counts the number of lattice points in integer dilates of a lattice polytope. A central question in Ehrhart theory is to characterize all possible Ehrhart polynomials. An important tool is the h*-polynomial of a lattice polytope, which encodes the Ehrhart polynomial in a certain binomial basis. One open question coming from commutative algebra is whether the h*-polynomial of an integrally closed lattice polytope is always unimodal. Schepers and Van Langenhoven (2011) proved this for lattice parallelepipeds. Using the interplay of geometry and combinatorics, we generalize their result to zonotopes by interpreting their h*-polynomials in terms of certain refined descent statistics on permutations. From that we obtain that the h*-polynomial of a zonotope is unimodal with peak in the middle and, moreover, that it has only real roots. This is joint work with Matthias Beck and Emily McCullough (both San Francisco State University).