Lattice polytopes with a given $h^*$-polynomial

Abstract

Let $Δ\subset \R^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_Δ^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |kΔ\cap \Z^n| t^k $ is a polynomial of degree $d \leq n$ with nonnegative integral coefficients. Let $AGL(n, \Z)$ be the group of invertible affine integral transformations which naturally acts on $\R^n$. For a given polynomial $h^* \in \Z[t]$, we denote by $C_{h^*}(n)$ the number $AGL(n, \Z)$-equivalence classes of $n$-dimensional lattice polytopes such that $h^* = h_Δ^*(t)$. In this paper we show that $\{C_{h^*}(n) \}_{n \geq 1}$ is a monotone increasing sequence which eventually becomes constant. This statement follows from a more general combinatorial result whose proof uses methods of commutative algebra.