Integer Points in Convex Simple Polytopes
by
Evgeni Materov
Krasnoyarsk State Technical University

Let Q be a convex polytope in Rⁿ with vertices in Zⁿ. The polytope Q is called simple if only n edges can meet in every vertex of Q. Given a positive integer k, write E(k, Q) for the number of integer points in k Q, and E^*(k, Q) for the number of integer points in the relative interior of polytope k Q. It is well-known that E(k, Q) is a polynomial of degree n on k, called the Erhart polynomial of Q. If k > 0 then

E(-k, Q) = (-1)n E*(k, Q).

The last formula is called the inversion formula. We introduce a new combinatorial object E_p(k, Q) - the p-th Erhart polynomial for which the generalized inversion formula holds true, i.e. for all 0 <= p <= n we have

Ep(-k, Q) = (-1)n En-p(k, Q).

For example, when p=0 we get the ordinary inversion formula. The proof of the formula above relies on some recent developments from the theory of toric varieties. As a corollary we get the relations between the numbers of integer points in convex simple polytopes.

Date received: February 17, 2000