Problem of the Week
Hilton Memorial Lecture
We study higher-dimensional analogs of the Dedekind-Carlitz polynomials,
c(u,v;a,b) := Sumk=1 .. a-1 uk-1 vfloor(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) = ua-1 vb-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. The talk will be accessible to a beginning graduate student.
This is joint work with Asia Matthews.