Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants

TL;DR

This paper introduces a new method using resultants to construct polylogarithmic motivic Chabauty-Kim functions for the punctured projective line, aiding solutions to the $S$-unit equation with explicit examples.

Contribution

It develops a novel approach to construct motivic Chabauty-Kim functions via resultants, improving the geometric step algorithm for solving Diophantine equations.

Findings

Constructed explicit motivic Chabauty-Kim function for $|S|=2$ in depth 6

Proved minimality of the constructed Chabauty-Kim function in degree and depth

Enhanced the geometric step algorithm for better efficiency

Abstract

Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach.