Title:  Integer-Point Transforms of Rational Polygons and Rademacher--Carlitz Polynomials  (joint work with Matthias Beck)

Abstract: We introduce and study the Rademacher--Carlitz polynomial. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view the Rademacher--Caritz polynomial as a polynomial analogue (in the sense of Carlitz) of the Dedekind--Rademacher sum which appears in various number-theoretic, combinatorial, geometric, and computational contexts.

 

Our results come in three flavors: we prove a reciprocity theorem for Rademacher--Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms ---which is a way of encoding integer points as a polynomial--- of any rational polyhedron P, and (if time allows) we derive a novel reciprocity theorem for Dedekind--Rademacher sums, which follows naturally from our setup.