On the functional equations for polylogarithms in one variable

TL;DR

This paper introduces a new method to analyze the functional equations of classical polylogarithms, providing a complete description for weight four polylogarithms and simplifying the understanding of these equations.

Contribution

The paper presents a novel approach inspired by Goncharov's conjectures, offering a complete characterization of functional equations for weight four polylogarithms in one variable.

Findings

Complete description of functional equations for weight four polylogarithm.

Simplified characterization of equations satisfied by dilogarithm and trilogarithm.

Sharpened version of Zagier's criterion for functional equations.

Abstract

We develop a new approach to the study of the functional equations satisfied by classical polylogarithms, inspired by Goncharov's conjectures. We prove a sharpened version of Zagier's criterion for such an equation and explain, how our approach leads to a very simple description of the equations in one variable, satisfied by dilogarithm and trilogarithm. Our main result is the complete description of the functional equations for weight four polylogarithm in one variable.