Generating Functions of Rational Polyhedra and Dedekind-Carlitz Polynomials

04/08/2008 - 12:15pm
04/08/2008 - 1:10pm
Matthias Beck (San Francisco State University)

We study higher-dimensional analogs of the Dedekind-Carlitz polynomials,
$ c(u,v;a,b) := \sum_{k=1}^{a-1} u^{k-1} v^{floor(kb/a)} $,
where $ u $ and $ v $ are indeterminates and a and b are positive integers. These polynomials satisfy the reciprocity law
$ (u-1) c(u,v;a,b) + (v-1) c(v,u;b,a) = u^{a-1}  v^{b-1} - 1  $,
from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations,
most notably by Hardy and Berndt-Dieter. Dedekind-Carlitz polynomials appear naturally in generating functions of
rational cones. We use this fact to give geometric proofs of the Carlitz reciprocity law. Our approach gives rise to
new reciprocity theorems and a multivariate generalization of the Mordell-Pommersheim theorem on the appearance
of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes. (I will not assume familiarity with Dedekind
sums or discrete geometry and I will carefully define all the terminology used above.) This is joint work with Christian
Haase (Freie Universit"at Berlin) and Asia Matthews (Queens University).

Millikan 208, Pomona College

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