{"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/separating-singular-moduli-and-the-primitive","title":"Separating singular moduli and the primitive element problem","arxiv_id":"1903.07126","date":"2019-03-17","proceeding":null,"authors":["Yuri Bilu","Bernadette Faye","Huilin Zhu"],"abstract":"We prove that $|x-y|\\ge 800X^{-4}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the \"primitive element problem\" for two singular moduli. In a previous article Faye and Riffaut show that the number field $\\mathbb Q(x,y)$, generated by two singular moduli $x$ and $y$, is generated by $x-y$ and, with some exceptions, by $x+y$ as well. In this article we fix a rational number $\\alpha \\ne0,\\pm1$ and show that the field $\\mathbb Q(x,y)$ is generated by $x+\\alpha y$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field over $\\mathbb Q$. Together with the above-mentioned result of Faye and Riffaut, this gives a drastic generalization of a theorem due to Allombert et al. (2015) about solution of linear equations in singular moduli.","url_abs":"http://arxiv.org/abs/1903.07126v4","url_pdf":"http://arxiv.org/pdf/1903.07126v4.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":"separating-singular-moduli-and-the-primitive","repo_url":"https://github.com/yuribilu/Separating","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}