[2404.02749] Universally defining subrings in function fields Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2404.02749 (math) [Submitted on 3 Apr 2024 ( v1 ), last revised 30 Apr 2026 (this version, v2)] Title: Universally defining subrings in function fields Authors: Nicolas Daans , Philip Dittmann View a PDF of the paper titled Universally defining subrings in function fields, by Nicolas Daans and 1 other authors View PDF HTML (experimental) Abstract: We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of $S$-integers is always a diophantine set. As a technical tool, we use a reciprocity exact sequence for quadratic Witt groups in function fields over almost arbitrary base fields (of any characteristic), which is new and of potentially independent interest. Comments: author accepted manuscript Subjects: Number Theory (math.NT) ; Logic (math.LO) MSC classes: 12L99 (primary), 11E81, 12L05, 12F20 (secondary) Cite as: arXiv:2404.02749 [math.NT] (or arXiv:2404.02749v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2404.02749 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J. Reine Angew. Math. 834 (2026), 229-270 Related DOI : https://doi.org/10.1515/crelle-2026-0026 Focus to learn more DOI(s) linking to related resources Submission history From: Nicolas Daans [ view email ] [v1] Wed, 3 Apr 2024 13:48:18 UTC (46 KB) [v2] Thu, 30 Apr 2026 11:50:01 UTC (50 KB) Full-text links: Access Paper: View a PDF of the paper titled Universally defining subrings in function fields, by Nicolas Daans and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2024-04 Change to browse by: math math.LO References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from