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Concept# Ehrhart polynomial

Summary

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.
Definition
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 \mathcal{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 \mathcal{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 o

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