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
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)