Boundary $h^\ast$-polynomials of rational polytopes

Abstract

If $P$ is a lattice polytope (i.e., $P$ is the convex hull of finitely many integer points in $\mathbb{R}^d$) of dimension $d$, Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|nP \cap \mathbb{Z}^d|$ is a degree-$d$ polynomial in the integer variable $n$. Equivalently, the generating function $1 + \sum_{n\geq 1} |nP \cap \mathbb{Z}^d| \, z^n$ is a rational function of the form $\frac{ h^\ast(z) }{ (1-z)^{ d+1 } }$; we call $h^\ast(z)$ the $h^\ast$-polynomial of $P$. There are several known necessary conditions for $h^\ast$-polynomials, including results by Hibi (1990), Stanley (1991), and Stapledon (2009), 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 $h^\ast$-polynomial of the boundary of a polytope, recovering all of the above results and their extensions for rational polytopes in a unifying manner. We include applications for (rational) Gorenstein polytopes and rational Ehrhart dilations.