Multidimensional Ehrhart Reciprocity

Details Multidimensional Ehrhart Reciprocity Multidimensional Ehrhart Reciprocity

2001-11-30

arxiv.org/abs/math/0111331v2

J. Combin. Th. Ser. A 97, no. 1 (2002), 187-194

Mathematics

Combinatorics

7 pages

Scientific paper

In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved that, if our polytope is a simplex, 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. In the present paper we complete the picture by extending this result to general rational polytopes. As a corollary, we also generalize a reciprocity theorem of Stanley.

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