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.
Definition.
Informally, if is a polytope, and is the polytope formed by expanding by a factor of in each dimension, then is the number of integer lattice points in .
More formally, consider a lattice formula_1 in Euclidean space formula_2 and a -dimensional polytope in formula_2 with the property that all vertices of the polytope are points of the lattice. (A common example is formula_4 and a polytope for which all vertices have integer coordinates.) For any positive integer , let be the -fold dilation of (the polytope formed by multiplying each vertex coordinate, in a basis for the lattice, by a factor of ), and let
be the number of lattice points contained in the polytope . Ehrhart showed in 1962 that is a rational polynomial of degree in , i.e. there exist rational numbers formula_6 such that:
for all positive integers .
The Ehrhart polynomial of the interior of a closed convex polytope can be computed as:
where is the dimension of . This result is known as Ehrhart–Macdonald reciprocity.
Examples.
Let be a -dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1. In terms of inequalities,
Then the -fold dilation of is a cube with side length , containing integer points. That is, the Ehrhart polynomial of the hypercube is . Additionally, if we evaluate at negative integers, then
as we would expect from Ehrhart–Macdonald reciprocity.
Many other figurate numbers can be expressed as Ehrhart polynomials. For instance, the square pyramidal numbers are given by the Ehrhart polynomials of a square pyramid with an integer unit square as its base and with height one; the Ehrhart polynomial in this case is .
Ehrhart quasi-polynomials.
Let be a rational polytope. In other words, suppose
where formula_12 and formula_13 (Equivalently, is the convex hull of finitely many points in formula_14) Then define
In this case, is a quasi-polynomial in . Just as with integral polytopes, Ehrhart–Macdonald reciprocity holds, that is,
Examples of Ehrhart quasi-polynomials.
Let be a polygon with vertices (0,0), (0,2), (1,1) and (, 0). The number of integer points in will be counted by the quasi-polynomial 
Interpretation of coefficients.
If is closed (i.e. the boundary faces belong to ), some of the coefficients of have an easy interpretation:
The Betke–Kneser theorem.
Ulrich Betke and Martin Kneser established the following characterization of the Ehrhart coefficients. A functional formula_21 defined on integral polytopes is an formula_22 and translation invariant valuation if and only if there are real numbers formula_23 such that
Ehrhart series.
We can define a generating function for the Ehrhart polynomial of an integral -dimensional polytope as
This series can be expressed as a rational function. Specifically, Ehrhart proved (1962) that there exist complex numbers formula_26, formula_27, such that the Ehrhart series of is
Additionally, Richard P. Stanley's non-negativity theorem states that under the given hypotheses, formula_26 will be non-negative integers, for formula_27.
Another result by Stanley shows that if is a lattice polytope contained in , then formula_31 for all . The formula_32-vector is in general not unimodal, but it is whenever it is symmetric, and the polytope has a regular unimodal triangulation.
Ehrhart series for rational polytopes.
As in the case of polytopes with integer vertices, one defines the Ehrhart series for a rational polytope. For a "d"-dimensional rational polytope , where is the smallest integer such that is an integer polytope ( is called the denominator of ), then one has
where the formula_26 are still non-negative integers.
Non-leading coefficient bounds.
The polynomial's non-leading coefficients formula_35 in the representation 
can be upper bounded:
where formula_38 is a Stirling number of the first kind. Lower bounds also exist.
Toric variety.
The case formula_39 and formula_40 of these statements yields Pick's theorem. 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 is the toric variety corresponding to the normal fan of , then defines an ample line bundle on , and the Ehrhart polynomial of coincides with the Hilbert polynomial of this line bundle.
Ehrhart polynomials can be studied for their own sake. For instance, one could ask questions related to the roots of an Ehrhart polynomial. Furthermore, some authors have pursued the question of how these polynomials could be classified.
Generalizations.
It is possible to study the number of integer points in a polytope if we dilate some facets of 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.
Counting the number of integer points in semi-dilations of polytopes has applications 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.