DedekindCarlitz Polynomials as LatticePoint Enumerators in Rational Polyhedra
Abstract
We study higherdimensional analogs of the DedekindCarlitz polynomials c(u,v;a,b) := sum_{k=1..b1} u^[ka/b] v^(k1), where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law (v1) c(u,v;a,b) + (u1) c(v,u;b,a) = u^(a1) v^(b1)  1, from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that DedekindCarlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for DedekindCarlitz polynomials, a characterization of DedekindCarlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the MordellPommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3dimensional lattice polytopes.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.1323
 Bibcode:
 2007arXiv0710.1323B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics;
 11P21;
 11L03;
 05A15;
 52C07
 EPrint:
 13 pages, 2 figures