04/08/2008 - 12:15pm

04/08/2008 - 1:10pm

Speaker:

Matthias Beck (San Francisco State University)

Abstract:

We study higher-dimensional analogs of the Dedekind-Carlitz polynomials,

,

where and are indeterminates and a and b are positive integers. These polynomials satisfy the reciprocity law

,

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).

Where:

Millikan 208, Pomona College

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