Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra

Mathematics – Number Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

13 pages, 2 figures

Scientific paper

We study higher-dimensional analogs of the Dedekind-Carlitz polynomials c(u,v;a,b) := sum_{k=1..b-1} u^[ka/b] 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) c(u,v;a,b) + (u-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. 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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-140367

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.