If P is a lattice polytope (i.e., P is the convex hull of finitely many integer points in R^d), Ehrhart's famous theorem asserts that the integer-point counting function |mP∩Z^d| is a polynomial in the integer variable m. Equivalently, the generating function \sum_{m \ge 0} |mP∩Z^d| t^m is a rational function of the form h*(t)/(1-t)^{d+1}; we call h*(t) the Ehrhart h*-polynomial of P. We know several necessary conditions for h*-polynomials, including results by Hibi, Stanley, and Stapledon, who used an interplay of arithmetic (integer-point structure) and topological (local h-vectors of triangulations) data of a given polytope. We introduce an alternative ansatz to understand Ehrhart theory through the h*-polynomial of the boundary of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner.

This is joint work with Esme Bajo (UC Berkeley).