2024/08/14 by Jarossay, David, Lilienfeldt, David T. -B. G., Saettone, Francesco Maria +2
#11D45 #11G55 #11Y50 #14G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2408.07400
Given a finite set S of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for ℙ1 ∖ \ 0,1,∞ \ using resultants. For a prime p\not∈ 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.