{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/effective-multiplicative-independence-of-3","title":"Effective multiplicative independence of 3 singular moduli","arxiv_id":"2207.05183","date":"2022-07-11","proceeding":null,"authors":["Yuri Bilu","Sanoli Gun","Emanuele Tron"],"abstract":"Pila and Tsimerman proved in 2017 that for every $k$ there exists at most finitely many $k$-tuples $(x_1,\\ldots, x_k)$ of distinct non-zero singular moduli with the property \"$x_1, \\ldots,x_k$ are multiplicatively dependent, but any proper subset of them is multiplicatively independent\". The proof was non-effective, using Siegel's lower bound for the Class Number. In 2019 Riffaut obtained an effective version of this result for $k=2$. Moreover, he determined all the instances of $x^my^n\\in \\mathbb Q^\\times$, where $x,y$ are distinct singular moduli and $m,n$ non-zero integers. In this article we obtain a similar result for $k=3$. We show that $x^my^nz^r\\in \\mathbb Q^\\times$ (where $x,y,z$ are distinct singular moduli and $m,n,r$ non-zero integers) implies that the discriminants of $x,y,z$ do not exceed $10^{10}$.","url_abs":"https://arxiv.org/abs/2207.05183v4","url_pdf":"https://arxiv.org/pdf/2207.05183v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"effective-multiplicative-independence-of-3","repo_url":"https://github.com/yuribilu/multiplicative","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}