Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants
David Jarossay, David T. -B. G. Lilienfeldt, Francesco Maria Saettone, Ariel Weiss, Sa'ar Zehavi
TL;DR
The paper develops a polylogarithmic motivic Chabauty–Kim framework for X = 𝑷^1 \ {0,1,∞} over the S-integers, separating the problem into a geometry-driven step and a p-adic arithmetic step. It introduces an algorithmic upper-bound method to count possible motivic CK functions and demonstrates that for |S|=2 there are no functions with depth- less than 6 and degree less than 18, while a first non-trivial function F^{|2|}_{6,18} exists at depth 6 and degree 18 built via a resultant construction. The main novelty is the geometric step via resultants, which yields an explicit, non-trivial CK function in depth 6; the function factors as a product involving depth-2 generators and a new depth-6 component, providing the first such explicit example for |S|≥2. Together with the dimension-based upper bounds, this work advances explicit Chabauty–Kim methods beyond depth 4 and paves the way toward handling multiple primes, though computational barriers remain for higher s. The combination of motivic structure, an elimination-by-resultants strategy, and explicit polynomial realizations offers a principled route to compute S-unit solutions via Chabauty–Kim theory in new regimes.
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.