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 introduced them in the 1960s.

Definition

Informally, if Template:Math is a polytope, and Template:Math is the polytope formed by expanding Template:Math by a factor of Template:Math in each dimension, then Template:Math is the number of integer lattice points in Template:Math.

More formally, consider a lattice <math>\mathcal{L}</math> in Euclidean space <math>\R^n</math> and a Template:Math-dimensional polytope Template:Math in <math>\R^n</math> with the property that all vertices of the polytope are points of the lattice. (A common example is <math>\mathcal{L} = \Z^n</math> and a polytope for which all vertices have integer coordinates.) For any positive integer Template:Math, let Template:Math be the Template:Math-fold dilation of Template:Math (the polytope formed by multiplying each vertex coordinate, in a basis for the lattice, by a factor of Template:Math), and let

<math>L(P,t) = \#\left(tP \cap \mathcal{L}\right)</math>

be the number of lattice points contained in the polytope Template:Math. Ehrhart showed in 1962 that Template:Math is a rational polynomial of degree Template:Math in Template:Math, i.e. there exist rational numbers <math>L_0(P),\dots,L_d(P)</math> such that:

<math>L(P, t) = L_d(P) t^d + L_{d-1}(P) t^{d-1} + \cdots + L_0(P)</math>

for all positive integers Template:Math.<ref name=ehrhart>Template:Citation</ref>

Reciprocity property

The Ehrhart polynomial of the interior of a closed convex polytope Template:Math can be computed as:

<math> L(\operatorname{int}(P), t) = (-1)^d L(P, -t),</math>

where Template:Math is the dimension of Template:Math. This result is known as Ehrhart–Macdonald reciprocity.<ref>Ehrhart, Eugène (1967), "Démonstration de la loi de réciprocité du polyèdre rationnel", Comptes Rendus de l'Academie des Sciences de Paris, Sér. A-B 265, A91–A94.</ref><ref>Template:Citation</ref>

Examples

File:Second dilate of a unit square.png

This is the second dilate, <math>t = 2</math>, of a unit square. It has nine integer points.

Let Template:Math be a Template:Math-dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1. In terms of inequalities,

<math> P = \left\{x\in\R^d : 0 \le x_i \le 1; 1 \le i \le d\right\}.</math>

Then the Template:Math-fold dilation of Template:Math is a cube with side length Template:Math, containing Template:Math integer points. That is, the Ehrhart polynomial of the hypercube is Template:Math.<ref>Template:Citation</ref><ref>Template:Citation</ref> Additionally, if we evaluate Template:Math at negative integers, then

<math>L(P, -t) = (-1)^d (t - 1)^d = (-1)^d L(\operatorname{int}(P), t),</math>

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 Template:Math.<ref>Template:Citation</ref>

Ehrhart quasi-polynomials

Let Template:Math be a rational polytope. In other words, suppose

<math>P = \left\{ x\in\R^d : Ax \le b\right\},</math>

where <math>A \in \Q^{k \times d}</math> and <math>b \in \Q^k.</math> (Equivalently, Template:Math is the convex hull of finitely many points in <math>\Q^d.</math>) Then define

<math>L(P, t) = \#\left(\left\{x\in\Z^d : Ax \le tb \right\} \right). </math>

In this case, Template:Math is a quasi-polynomial in Template:Math. Just as with integral polytopes, Ehrhart–Macdonald reciprocity holds, that is,

<math> L(\operatorname{int}(P), t) = (-1)^d L(P, -t). </math>

Examples of Ehrhart quasi-polynomials

Let Template:Math be a polygon with vertices (0,0), (0,2), (1,1) and (Template:Sfrac, 0). The number of integer points in Template:Math will be counted by the quasi-polynomial <ref name=MR2271992>Template:Citation</ref>

<math> L(P, t) = \frac{7t^2}{4} + \frac{5t}{2} + \frac{7 + (-1)^t}{8}. </math>

Interpretation of coefficients

If Template:Math is closed (i.e. the boundary faces belong to Template:Math), some of the coefficients of Template:Math have an easy interpretation:

• the leading coefficient, <math>L_d(P)</math>, is equal to the Template:Math-dimensional volume of Template:Math, divided by Template:Math (see lattice for an explanation of the content or covolume Template:Math of a lattice);

• the second coefficient, <math>L_{d-1}(P)</math>, can be computed as follows: the lattice Template:Math induces a lattice Template:Math on any face Template:Math of Template:Math; take the Template:Math-dimensional volume of Template:Math, divide by Template:Math, and add those numbers for all faces of Template:Math;

• the constant coefficient, <math>L_0(P)</math>, is the Euler characteristic of Template:Math. When Template:Math is a closed convex polytope, <math>L_0(P)=1.</math>

The Betke–Kneser theorem

Ulrich Betke and Martin Kneser<ref>Template:Citation</ref> established the following characterization of the Ehrhart coefficients. A functional <math>Z</math> defined on integral polytopes is an <math>\operatorname{SL}(n,\Z)</math> and translation invariant valuation if and only if there are real numbers <math>c_0,\ldots, c_n</math> such that

<math> Z= c_0 L_0+\cdots +c_n L_n.</math>

Ehrhart series

We can define a generating function for the Ehrhart polynomial of an integral Template:Math-dimensional polytope Template:Math as

<math> \operatorname{Ehr}_P(z) = \sum_{t\ge 0} L(P, t)z^t. </math>

This series can be expressed as a rational function. Specifically, Ehrhart proved (1962) that there exist complex numbers <math>h_j^*</math>, <math>0 \le j \le d</math>, such that the Ehrhart series of Template:Math is<ref name=ehrhart/>

<math>\operatorname{Ehr}_P(z) = \frac{\sum_{j=0}^d h_j^\ast(P) z^j}{(1 - z)^{d + 1}}, \qquad \sum_{j=0}^d h_j^\ast(P) \neq 0.</math>

Richard P. Stanley's non-negativity theorem states that under the given hypotheses, <math>h_j^*</math> will be non-negative integers, for <math>0 \le j \le d</math>.

Another result by Stanley shows that if Template:Math is a lattice polytope contained in Template:Math, then <math>h_j^*(P) \le h_j^*(Q)</math> for all Template:Math.<ref>Template:Citation</ref> The <math>h^*</math>-vector is in general not unimodal, but it is whenever it is symmetric and the polytope has a regular unimodular triangulation.<ref>Template:Citation</ref>

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 Template:Math, where Template:Math is the smallest integer such that Template:Math is an integer polytope (Template:Math is called the denominator of Template:Math), then one has

<math>\operatorname{Ehr}_P(z) = \sum_{t\ge 0} L(P, t)z^t = \frac{\sum_{j=0}^{D(d+1)} h_j^\ast(P) z^j}{\left(1 - z^D\right)^{d + 1}},</math>

where the <math>h_j^*</math> are still non-negative integers.<ref>Template:Citation</ref><ref>Template:Citation</ref>

Non-leading coefficient bounds

The polynomial's non-leading coefficients <math>c_0,\dots,c_{d-1}</math> in the representation

<math>L(P,t) = \sum_{r=0}^d c_r t^r</math>

can be upper bounded:<ref>Template:Citation</ref>

<math>c_r \leq |s(d,r)| c_d + \frac{1}{(d-1)!} |s(d,r+1)|</math>

where <math>s(n,k)</math> is the Stirling number of the first kind. Lower bounds also exist.<ref>Template:Citation</ref>

Toric variety

The case <math>n=d=2</math> and <math>t = 1</math> 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 Template:Math is the toric variety corresponding to the normal fan of Template:Math, then Template:Math defines an ample line bundle on Template:Math, and the Ehrhart polynomial of Template:Math 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.<ref>Template:Citation</ref> Furthermore, some authors have pursued the question of how these polynomials could be classified.<ref>Template:Citation</ref>

Generalizations

It is possible to study the number of integer points in a polytope Template:Math if we dilate some facets of Template:Math 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.<ref>Template:Citation</ref>

Counting the number of integer points in semi-dilations of polytopes has applications<ref>Template:Citation</ref> 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.

See also

• Quasi-polynomial
• Stanley's reciprocity theorem

Further reading

• Template:Citation. Introduces the Fourier analysis approach and gives references to other related articles.
• Template:Citation.