Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants
| Autor: | Jarossay, David, Lilienfeldt, David T. -B. G., Saettone, Francesco Maria, Weiss, Ariel, Zehavi, Sa'ar |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 31 pages, code available at GitHub repository, comments welcome |
| Databáze: | arXiv |
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