Dedekind–Carlitz polynomials as lattice-point enumerators in rational polyhedra

Abstract

We study higher-dimensional analogs of the Dedekind–Carlitz polynomials $${c(u,v;a,b): = \sum\limits_{k = 1}^{b - 1} {u^{\left[ {\frac{{ka}}{b}} \right]} } v^{k - 1}}$$ , where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law $${(v - 1)\,{\rm c}\,(u, v; a, b) + (u - 1)\,{\rm c}\,(v, u; b, a) = u^{a-1} v^{b-1} - 1,}$$ from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind–Carlitz 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 Dedekind–Carlitz polynomials, a characterization of Dedekind–Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell–Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes.