Lattice polytopes with a given $h^{\ast }$-polynomial

Abstract

Let $\Delta \subset {\mathbb{R}}^n$ be an $n$-dimensional lattice polytope. It is well-known that $h_{\Delta }^{\ast }(t):=(1-t{)}^{n+1}{\sum }_{k\ge 0}|k\Delta \cap {\mathbb{Z}}^n|t^k$ is a polynomial of degree $d\le n$ with nonnegative integral coefficients. Let $AGL(n,\mathbb{Z})$ be the group of invertible affine integral transformations which naturally acts on ${\mathbb{R}}^n$. For a given polynomial $h^{\ast }\in \mathbb{Z}[t]$, we denote by $C_{h^{\ast }}(n)$ the number $AGL(n,\mathbb{Z})$-equivalence classes of $n$-dimensional lattice polytopes such that $h^{\ast }=h_{\Delta }^{\ast }(t)$. In this paper we show that $\{C_{h^{\ast }}(n){\}}_{n\ge 1}$ 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.

10 pages. AMS-LaTeX, some typos were corrected