[Paper Review] Dedekind sums: a combinatorial-geometric viewpoint

Matthias Beck, Sinai Robins|ArXiv.org|Dec 7, 2001

Advanced Mathematical Theories3 citations

TL;DR

This paper introduces Fourier-Dedekind sums as a unified combinatorial-geometric framework for generalized Dedekind sums, showing they naturally arise in lattice point enumeration of rational polytopes and partition functions. Using generating functions and Barvinok's algorithm, the authors prove that Zagier's higher-dimensional Dedekind sums are computable in polynomial time for fixed dimension, generalizing classical reciprocity laws and unifying diverse number-theoretic and geometric results.

ABSTRACT

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers. This problem also goes by the name of the `coin exchange problem'. Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topology, number theory, and combinatorial geometry. The Fourier-Dedekind sums we study here include as special cases generalized Dedekind sums studied by Berndt, Carlitz, Grosswald, Knuth, Rademacher, and Zagier. Our interest in these sums stems from the appearance of Dedekind's and Zagier's sums in lattice point count formulas for polytopes. Using some simple generating functions, we show that generalized Dedekind sums are natural ingredients for such formulas. As immediate `geometric' corollaries to our formulas, we obtain and generalize reciprocity laws of Dedekind, Zagier, and Gessel. Finally, we prove a polynomial-time complexity result for Zagier's higher-dimensional Dedekind sums.

Motivation & Objective

• To unify diverse generalizations of Dedekind sums through a combinatorial-geometric framework rooted in lattice point enumeration of rational polytopes.
• To establish that Fourier-Dedekind sums serve as fundamental building blocks for the partition function of integers from a finite set of positive integers.
• To derive and generalize reciprocity laws of Dedekind, Zagier, and Gessel as geometric consequences of lattice point generating functions.
• To prove that Zagier's higher-dimensional Dedekind sums are computable in polynomial time for fixed dimension, resolving a key complexity question.

Proposed method

• Defining Fourier-Dedekind sums as finite Fourier series over roots of unity: $\sigma_n(a_1,\dots,a_d;a_0) = \frac{1}{a_0}\sum_{\lambda^{a_0}=1} \frac{\lambda^n}{(1-\lambda^{a_1})\cdots(1-\lambda^{a_d})}$, excluding singular terms.
• Using generating functions for rational polytopes to express the number of integer partitions as a quasipolynomial whose coefficients are built from Fourier-Dedekind sums.
• Applying Barvinok's theorem on rational generating functions to show that the generating function of a rational cone can be computed in polynomial time for fixed dimension.
• Deriving reciprocity laws by analyzing the structure of generating functions and their behavior under duality, generalizing classical results.
• Proving polynomial-time computability of higher-dimensional Dedekind sums via induction and the virtual decomposition of rational generating functions.
• Connecting the residue of a meromorphic function in the main theorem of [DR] to the sum of cotangents, which yields Zagier's sums.

Experimental results

Research questions

• RQ1 How can generalized Dedekind sums be systematically unified through a common combinatorial-geometric framework?
• RQ2 In what way do Fourier-Dedekind sums naturally arise in the enumeration of lattice points in rational polytopes and integer partition functions?
• RQ3 Can classical reciprocity laws for Dedekind and Zagier sums be derived as geometric consequences of generating function identities?
• RQ4 Is Zagier's higher-dimensional Dedekind sum computable in polynomial time for fixed dimension, and if so, what algorithmic tools enable this?
• RQ5 What is the role of generating functions and virtual decompositions in establishing complexity bounds for lattice point counting and related number-theoretic functions?

Key findings

• Fourier-Dedekind sums provide a unifying framework for generalized Dedekind sums, including those of Berndt, Carlitz, Grosswald, Knuth, Rademacher, and Zagier.
• The number of integer partitions of $n$ from a finite set of positive integers is a quasipolynomial whose coefficients are built from Fourier-Dedekind sums.
• Reciprocity laws of Dedekind, Zagier, and Gessel are derived as geometric consequences of lattice point generating function identities.
• Zagier's higher-dimensional Dedekind sums are polynomial-time computable in fixed dimension, proven via Barvinok's algorithm and induction on dimension.
• The generating function of a rational cone admits a virtual decomposition into rational functions with polynomial-time computable coefficients, enabling efficient computation of lattice point counts.
• The sum $\sum_{k=1}^{a_0-1} \cot(\pi k a_1 / a_0) \cdots \cot(\pi k a_d / a_0)$, which defines the higher-dimensional Dedekind sum, is polynomial-time computable for fixed $d$.