Ehrhart polynomial
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.
with the property that all vertices of the polytope are points of the lattice.
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.
such that: for all positive integers t.[1] The Ehrhart polynomial of the interior of a closed convex polytope P can be computed as: where d is the dimension of P. This result is known as Ehrhart–Macdonald reciprocity.
[2] Let P be a d-dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1.
In terms of inequalities, Then the t-fold dilation of P is a cube with side length t, containing (t + 1)d integer points.
That is, the Ehrhart polynomial of the hypercube is L(P,t) = (t + 1)d.[3][4] Additionally, if we evaluate L(P, t) at negative integers, then as we would expect from Ehrhart–Macdonald reciprocity.
Many other figurate numbers can be expressed as Ehrhart polynomials.
(Equivalently, P is the convex hull of finitely many points in
) Then define In this case, L(P, t) is a quasi-polynomial in t. Just as with integral polytopes, Ehrhart–Macdonald reciprocity holds, that is, Let P be a polygon with vertices (0,0), (0,2), (1,1) and (3/2, 0).
The number of integer points in tP will be counted by the quasi-polynomial [6] If P is closed (i.e. the boundary faces belong to P), some of the coefficients of L(P, t) have an easy interpretation: Ulrich Betke and Martin Kneser[7] established the following characterization of the Ehrhart coefficients.
Specifically, Ehrhart proved (1962) that there exist complex numbers
, such that the Ehrhart series of P is[1] Richard P. Stanley's non-negativity theorem states that under the given hypotheses,
Another result by Stanley shows that if P is a lattice polytope contained in Q, then
-vector is in general not unimodal, but it is whenever it is symmetric and the polytope has a regular unimodular triangulation.
Formulas for the other coefficients are much harder to get; Todd classes of toric varieties, the Riemann–Roch theorem as well as Fourier analysis have been used for this purpose.
If X is the toric variety corresponding to the normal fan of P, then P defines an ample line bundle on X, and the Ehrhart polynomial of P coincides with the Hilbert polynomial of this line bundle.
For instance, one could ask questions related to the roots of an Ehrhart polynomial.
[14] Furthermore, some authors have pursued the question of how these polynomials could be classified.
[15] It is possible to study the number of integer points in a polytope P if we dilate some facets of P but not others.
In other words, one would like to know the number of integer points in semi-dilated polytopes.
It turns out that such a counting function will be what is called a multivariate quasi-polynomial.
An Ehrhart-type reciprocity theorem will also hold for such a counting function.
[16] Counting the number of integer points in semi-dilations of polytopes has applications[17] in enumerating the number of different dissections of regular polygons and the number of non-isomorphic unrestricted codes, a particular kind of code in the field of coding theory.
