Describing all multivariable functional equations of dilogarithms

TL;DR

This paper provides a comprehensive characterization of all multivariable functional equations for dilogarithms, establishing their derivation from fundamental 5-term and 2-term relations, with detailed considerations for various dilogarithm variants.

Contribution

It offers the first complete description of multivariable dilogarithm equations, including arbitrary coefficients, torsion, and branch independence, unifying several existing results.

Findings

All multivariable dilogarithm equations follow from basic 5-term and 2-term relations.

The results hold for Bloch-Wigner, Rogers, and Coleman dilogarithms with arbitrary coefficients.

The paper establishes branch independence for Coleman's p-adic dilogarithm.

Abstract

We prove quite general statements about functional equations in any number of variables for the dilogarithms defined by Bloch-Wigner, Rogers, and Coleman, showing that they follow from certain 5-term and 2-term relations in a precise way. Unlike many other references, we use arbitrary coefficients, do not ignore any torsion, and get sharp results. For the Bloch-Wigner dilogarithm, we also consider complex conjugation, and for Coleman's p-adic dilogarithm we show independence of the branch.