A Closer Look at Lattice Points in Rational Simplices

Details A Closer Look at Lattice Points in Rational Simplices A Closer Look at Lattice Points in Rational Simplices

2003-06-02

arxiv.org/abs/math/0306034v1

Electronic J. Comb. 6, no. 1 (1999), R 37

Mathematics

Combinatorics

9 pages

Scientific paper

We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we use its description as the intersection of n+1 halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a "vector-dilated" simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon.

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