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

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 $p\notin S$, 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 $|S|=2$, 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.

arXiv commentv3: Accepted Manuscript, to appear in Algebra & Number Theory