# Ehrhart polynomial

In mathematics, integral polytopes have associated **Ehrhart polynomials** which encode some geometrical information about them.

Specifically, consider a lattice *L* in Euclidean space **R**^{n} and an *n*-dimensional polytope *P* in **R**^{n}, and assume that all vertices of the polytope are points of the lattice. (A common example is *L* = **Z**^{n} and a polytope with all its vertex coordinates being integers.) For any positive integer *t*, let *tP* be the *t*-fold dilation of *P* and let *L*(*P*, *t*) be the number of lattice points contained in *tP*. Ehrhart showed in 1967 that *L* is a rational polynomial of degree *n* in *t*, i.e. there exist rational numbers *a*_{0},...,*a*_{n} such that:

*L*(*P*,*t*) =*a*_{n}*t*^{n}+*a*_{n-1}*t*^{n-1}+ ... +*a*_{0}for all positive integers*t*.

Furthermore, if *P* is closed (i.e. the boundary faces belong to *P*), some of the coefficients of *L*(*P*, *t*) have an easy interpretation:

- the leading coefficient,
*a*_{n}, is equal to the*n*-dimensional volume of*P*, divided by*d*(*L*) (see lattice for an explanation of the content*d*(*L*) of a lattice); - the second coefficient,
*a*_{n-1}, can be computed as follows: the lattice*L*induces a lattice*L*on any face_{F}*F*of*P*; take the (*n*-1)-dimensional volume of*F*, divide by 2*d*(*L*), and add those numbers for all faces of_{F}*P*; - the constant coefficient
*a*_{0}is the Euler characteristic of*P*.

The case *n*=2 and *t*=1 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.

The Ehrhart polynomial of the interior of a closed polytope *P* can be computed as:

*L*(int*P*,*t*) = (-1)^{n}*L*(*P*, −*t*).

## References

- E. Ehrhart:
*Sur un problème géométrie diophantienne linéaire II*, Journal Reine Angewandte Mathematik 227 (1967), pp. 25-49. Definition and first properties. - Ricardo Diaz, Sinai Robins:
*The Ehrhart polynomial of a lattice*n*-simplex*, Electronic Research Announcements of the American Mathematical Society 2 (1996), pages 1-6, online version. Introduces the Fourier analysis approach and gives references to other related articles.