On Stanley's reciprocity theorem for rational cones

2004/09/28 by Matthias Beck, Beck, Matthias, Mike Develin +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #History and Theory of Mathematics #math.CO #msc:05A15 #msc:52C07 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0409562

9 pages

arxiv created 2005/08/04 · arxiv updated 2009/12/01

Abstract

We give a short, self-contained proof of Stanley's reciprocity theorem for a rational cone K ⊂ R^d. Namely, let sigma_K (x) = sum_{m ∈ K ∩ Z^d} x^m. Then sigma_K (x) and sigma_int(K) (x) are rational functions which satisfy the identity sigma_K (1/x) = (-1)^d sigma_int(K) (x). A corollary of Stanley's theorem is the Ehrhart-Macdonald reciprocity theorem for the lattice-point enumerator of rational polytopes. A distinguishing feature of our proof is that it uses neither the shelling of a polyhedron nor the concept of finite additive measures. The proof follows from elementary techniques in contour integration.

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