# The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory I

## TL;DR

This paper advances computational methods in Kim's conjecture for the thrice punctured line by refining algorithms and focusing on the polylogarithmic quotient, enabling verification of new cases.

## Contribution

It develops a refined algorithm tailored to the polylogarithmic quotient, enhancing explicit motivic computations in Kim's method for the thrice punctured line.

## Key findings

- Verified a new case of Kim's conjecture
- Developed a refined algorithm for motivic iterated integrals
- Highlighted symmetry-breaking effects in the polylogarithmic quotient

## Abstract

Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of $\mathrm{Spec}\,\mathbb{Z}$. To do so, we develop a greatly refined version of the algorithm of Dan-Cohen tailored specifically to this case, and we focus attention on the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. We also discover an interesting consequence of the symmetry-breaking nature of the polylog quotient that forces us to symmetrize our polylogarithmic version of Kim's conjecture. In this first part of a two-part series, we focus on a specific example, which allows us to verify an interesting new case of Kim's conjecture.