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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]

The normal cone of a face $F$ of a polytope consists of linear functionals whose minimum is attained on all of $F$. These cones form the inward normal fan, complete in the dual of the polytope's affine span. For a full-dimensional lattice polytope, it is a complete rational fan in the full dual space.

// Target: geometry-and-topology.bigb

For a lattice polytope $\Delta$, form the semigroup of lattice points $(m,l)$ with $l\ge 0$ and $m\in l\Delta$, using only $(0,0)$ in degree zero. Its semigroup algebra is graded by $l$, and its Proj construction gives ${\mathbb{P}}_{\Delta }$. The chart at a lattice vertex $v$ is $\mathrm{Spec}k[\mathrm{cone}(\Delta -v)\cap M]$; these charts give the normal fan of a polytope construction. For a polytope of smaller dimension, use its intrinsic affine lattice after translation.

// Target: geometry-and-topology.bigb