Counting Lattice Points by Means of the Residue Theorem

Author: Beck, M.

Source: The Ramanujan Journal, Volume 4, Number 3, 1 September 2000 , pp. 299-310(12)

Publisher: Springer

Buy & download fulltext article:

Price: $47.00 plus tax (Refund Policy)

Abstract:

We use the residue theorem to derive an expression for the number of lattice points in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be used to compute the Ehrhart polynomial for an arbitrary convex lattice polytope.

Keywords: lattice point enumeration in polytopes; Ehrhart polynomial; Dedekind sums

Document Type: Regular paper

Affiliations: 1: Department of Mathematical Sciences, State University of New York, Binghamton, New York 13902-6000. matthias@math.binghamton.edu

Publication date: 2000-09-01