  Let $\Delta \subset \R^n$ be an $n$-dimensional lattice polytope. It is
well-known that $h_{\Delta}^*(t) := (1-t)^{n+1} \sum_{k \geq 0} |k\Delta \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 $h^* = h_{\Delta}^*(t)$. 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.
