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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]

Ehrhart's theorem says that the lattice-point count $L_P(n)$ of a $d$-dimensional lattice polytope is a polynomial in the nonnegative integer dilation factor, of degree $d$. For full-dimensional $P\subset {\mathbb{R}}^d$, its leading coefficient is the ordinary volume and its constant coefficient is one. Rational vertices instead generally produce a quasipolynomial.^[ref]