Definition
Informally, if P is any polyhedron or polytope, and tP is the polytope formed by expanding P by a factor of t in each dimension, then L(int P, t) is the number of integer lattice points in tP.
More formally, consider a lattice L in Euclidean space Rn and a d-dimensional polytope P in Rn, and assume that all the vertices of the polytope are points of the lattice. (A common example is L = Zn and a polytope with all its vertex coordinates being integers.) 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 tP. Ehrhart showed in 1962 that L is a rational polynomial of degree d in t, i.e. there exist rational numbers a0,...,ad such that:
- L(P, t) = adtd + ad−1td−1 + … + a0 for all positive integers t.
The Ehrhart polynomial of the interior of a closed convex polytope P can be computed as:
- L(int P, t) = (−1)n L(P, −t).
Read more about this topic: Ehrhart Polynomial
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