Ehrhart polynomial

In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.

These polynomials are named after Eugène Ehrhart who studied them in the 1960s.

Informally, if $P$ is a polytope, and $tP$ is the polytope formed by expanding $P$ by a factor of $t$ in each dimension, then $L (P, t)$ is the number of integer lattice points in $tP$ .

More formally, consider a lattice $L$ in Euclidean space $ℝ n$ and a $d$ -dimensional polytope $P$ in $ℝ n$ with the property that all vertices of the polytope are points of the lattice. (A common example is $L = ℤ n$ and a polytope for which all vertices have integer coordinates.) For any positive integer $t$ , let $tP$ be the $t$ -fold dilation of $P$ (the polytope formed by multiplying each vertex coordinate, in a basis for the lattice, by a factor of $t$ ), and let

be the number of lattice points contained in the polytope $tP$ . Ehrhart showed in 1962 that $L$ is a rational polynomial of degree $d$ in $t$ , i.e. there exist rational numbers $a 0,..., a d$ such that:

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