On the Goncharov Depth Conjecture and polylogarithms of depth two

TL;DR

This paper proves the surjectivity of Goncharov's depth conjecture and demonstrates that multiple polylogarithms of depth two can be expressed using a specific single function, advancing understanding of polylogarithm structures.

Contribution

It establishes the surjectivity of Goncharov's depth conjecture and proves a key expression for depth two polylogarithms, linking them to a single polylogarithm function.

Findings

Proves surjectivity of Goncharov's depth conjecture

Shows depth d polylogarithms can be expressed via a single function for certain cases

Validates the expression for depth two polylogarithms

Abstract

We prove the surjectivity part of Goncharov's depth conjecture. We also show that the depth conjecture implies that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function $\mathrm{Li}_{n-d+1,1,\dots,1}(a_1,a_2,\dots,a_d)$, and we prove this latter statement for $d=2$.