Ehrhart polynomial

Summary

The Ehrhart polynomial of a lattice polytope $P$ counts the integer points in the dilates $tP$ for positive integers $t$. Ehrhart's theorem says that this count is a polynomial in $t$ of degree $\dim P$. Its leading coefficient is the volume of $P$ when $P$ is full-dimensional, and its constant term is $1$. It is a discrete analogue of volume, and it satisfies Ehrhart–Macdonald reciprocity.