{"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/some-uniform-effective-results-on-andre-oort","title":"Some uniform effective results on André--Oort for sums of powers in $\\mathbb{C}^n$","arxiv_id":"2405.06456","date":"2024-05-10","proceeding":null,"authors":["Guy Fowler"],"abstract":"We prove an Andr\\'e--Oort-type result for a family of hypersurfaces in $\\mathbb{C}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for the class number. We prove that, for $m, n \\in \\mathbb{Z}_{>0}$, there exists an effective constant $c(m, n)>0$ with the following property: if pairwise distinct singular moduli $x_1, \\ldots, x_n$ with respective discriminants $\\Delta_1, \\ldots, \\Delta_n$ are such that $a_1 x_1^m + \\ldots + a_n x_n^m \\in \\mathbb{Q}$ for some $a_1, \\ldots, a_n \\in \\mathbb{Q} \\setminus \\{0\\}$ and $\\# \\{ \\Delta_i : \\mathbb{Q}(\\sqrt{\\Delta_i}) = K_*\\} \\leq 1$, then $\\max_i \\lvert \\Delta_i \\rvert \\leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \\in \\mathbb{Q}$ for some $a_1, a_2, a_3 \\in \\mathbb{Q} \\setminus \\{0\\}$.","url_abs":"https://arxiv.org/abs/2405.06456v1","url_pdf":"https://arxiv.org/pdf/2405.06456v1.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":"some-uniform-effective-results-on-andre-oort","repo_url":"https://github.com/guyfowler/sums_of_powers","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}