Counting Lattice Points by means of the Residue Theorem

Details Counting Lattice Points by means of the Residue Theorem Counting Lattice Points by means of the Residue Theorem

2003-06-02

arxiv.org/abs/math/0306035v1

Ramanujan J. 4, no. 3 (2000), 299-310

Mathematics

Combinatorics

14 pages

Scientific paper

We use the residue theorem to derive an expression for the number of lattice oints 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.

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