Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7

TL;DR

This paper introduces new functional equations for multiple polylogarithms at weights 5, 6, and 7, enabling explicit depth reduction and generalizing key identities used in proving Zagier's Polylogarithm Conjecture.

Contribution

It provides novel functional equations for higher weights that extend previous identities, facilitating depth reduction of multiple polylogarithms.

Findings

New functional equations for weights 5, 6, and 7

Explicit depth reduction formulas for multiple polylogarithms

Generalization of the identity Q_4 used in Zagier's conjecture proof

Abstract

We present new functional equations in weights 5, 6 and 7 and use them for explicit depth reduction of multiple polylogarithms. These identities generalize the crucial identity $\mathbf{Q}_4$ from the recent work of Goncharov and Rudenko that was used in their proof of the weight 4 case of Zagier's Polylogarithm Conjecture.