Sur la conjecture de Zagier pour n=4. II

TL;DR

This paper advances the understanding of Zagier's conjecture for n=4 by expressing complex multiple polylogarithms as explicit linear combinations of simpler forms, leading to new functional equations.

Contribution

It provides explicit formulas for multiple polylogarithms of weight 4 in terms of (3,1) type polylogarithms, and derives a novel four-parameter functional equation.

Findings

Explicit linear combination formulas for weight 4 polylogarithms

New functional equation involving (3,1) polylogarithms

Progress towards Zagier's conjecture for n=4

Abstract

We express a general multiple polylogarithm of weight n as an explicit linear combination of multiple polylogarithms of weight n in n-2 variables. We express a general multiple polylogarithm of weight 4 as an explicit linear combination of multiple polylogarithms of type (3,1). We deduce a 4 parameters functional equation expressing a certain linear combination of multiple polylogarithms of type (3,1) as a linear combination of polylogarithms of weight 4.