Weighted Ehrhart Theory

In 1962 Ehrhart proved that the number of lattice points in integer dilates of a lattice polytope is given by a polynomial — the Ehrhart polynomial of the polytope. Since then Ehrhart theory has developed into a very active area of research at the intersection of combinatorics, geometry and algebra.

The Ehrhart polynomial encodes important information about the polytope such as its volume and the dimension. An important tool to study Ehrhart polynomials is the $h*-polynomial$, a linear transform of the Ehrhart polynomial which is given by the numerator of the generating series. By a famous theorem of Stanley the coefficients of the $h*-polynomial$ are always nonnegative integers. In this talk, we discuss generalizations of this result to weighted lattice point enumeration in rational polytopes where the weight function is given by a polynomial. In particular, we show that Stanley’s Nonnegativity Theorem continues to hold if the weight is a sum of products of linear forms that a nonnegative over the polytope.

This is joint work with Esme Bajo, Robert Davis, Jesús De Loera, Alexey Garber, Sofía Garzón Mora and Josephine Yu.