Papers › Polylogarithmic motivic Chabauty-Kim for ℙ¹ ∖{ 0,1,∞}: the geometric step via resultants
Polylogarithmic motivic Chabauty-Kim for ℙ¹ ∖{ 0,1,∞}: the geometric step via resultants
David Jarossay, David T. -B. G. Lilienfeldt, Francesco Maria Saettone, Ariel Weiss, Sa'ar Zehavi
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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∉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.
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