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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]

For a $d$-dimensional lattice polytope $P$ and positive integer $n$, Ehrhart reciprocity stateswith relative interior when needed. Negative input means polynomial evaluation, not a negative dilation count. For the unit square, $(1-n{)}^2=(n-1{)}^2$ counts the interior grid points of its $n$th dilation.^[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]

Matthias Beck coauthored a text on integer-point enumeration in polyhedra with Sinai Robins. It connects generating functions, Ehrhart polynomials and Ehrhart reciprocity. Counting lattice points in dilated shapes thereby reveals geometric information such as volume, while reciprocity relates counts in the interior to counts in the closed shape.^[ref]

Sinai Robins coauthored a text on integer-point enumeration with Matthias Beck. Their exposition develops Ehrhart polynomials, generating functions and Ehrhart reciprocity for polyhedra. The subject gives a precise bridge between discrete counting and continuous geometry: the leading growth of lattice-point counts records volume.^[ref]