Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem

[2608.15119] Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2608.15119 (math) [Submitted on 15 Aug 2026] Title: Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem Authors: Federico Accossato , Nadir Murru , Giuliano Romeo , Giulia Salvatori View a PDF of the paper titled Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem, by Federico Accossato and 3 other authors View PDF HTML (experimental) Abstract: Given a field $K$, let $K((T^{-1}))$ be the field of formal power series. Continued fractions in $K((T^{-1}))$ can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of $K((T^{-1}))$ arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that $K$ has characteristic zero, we improve the known analogues of the Maillet--Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport--Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy. Comments: 20 pages, comments are welcome! Subjects: Number Theory (math.NT) MSC classes: 11J70, 11D88, 11J81 Cite as: arXiv:2608.15119 [math.NT] (or arXiv:2608.15119v1 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.15119 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Giuliano Romeo [ view email ] [v1] Sat, 15 Aug 2026 08:40:44 UTC (18 KB) Full-text links: Access Paper: View a PDF of the paper titled Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem, by Federico Accossato and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-08 Change to browse by: math 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

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