# Lecture by Matthias Beck (San Francisco State University): Boundary h*-polynomials of rational polytopes

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).

### Time & Location

Jun 13, 2022 | 02:15 PM

Technische Universität Berlin

Institut für Mathematik

Straße des 17. Juni 136

10623 Berlin

Room MA 041 (Ground Floor)