 Ehrhart polynomial

For a lattice polytope $P$, its Ehrhart polynomial counts integer points in dilates: $L_P(n)=\#(nP\cap {\mathbb{Z}}^d)$ for nonnegative integers $n$. Ehrhart's theorem makes this counting function a polynomial. For the unit square, $L_P(n)=(n+1{)}^2$: there are $n+1$ choices for each coordinate. This turns a geometric counting problem into algebra.^[ref]

The Ehrhart polynomial is a mathematical tool used in the field of combinatorial geometry, particularly in the study of polytopes and their integer points. Specifically, it counts the number of integer points in the integer dilations of a rational polytope.