Bialgebraic geometry of Böttcher coordinates

[2606.24553] Bialgebraic geometry of Böttcher coordinates Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2606.24553 (math) [Submitted on 23 Jun 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: Bialgebraic geometry of Böttcher coordinates Authors: Sina Saleh View PDF Abstract: Becker and Bergweiler showed that if $f$ is a non-exceptional polynomial, then the Böttcher coordinate $\Psi_f \colon \mathbb D_R \to B_\infty(f)$ associated to $f$ is a transcendental function. In this paper, we study $f$-bialgebraic sets: algebraic subsets of $\mathbb D_R^n$ whose image under the coordinate-wise action of $\Psi_f$ is contained in an algebraic set of the same dimension. We give a complete dynamical classification of bialgebraic sets under the additional assumption that the Julia set of $f$ is either disconnected, or connected and admits a nondegenerate locally connected model. Inspired by the Ax--Lindemann--Weierstrass theorem and the Ax--Schanuel conjecture, we formulate analogs with $\Psi_f$ in place of the exponential function and prove them in the case where the Julia set $J_f$ is disconnected. Subjects: Dynamical Systems (math.DS) ; Number Theory (math.NT) Cite as: arXiv:2606.24553 [math.DS] (or arXiv:2606.24553v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2606.24553 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sina Saleh [ view email ] [v1] Tue, 23 Jun 2026 13:20:35 UTC (3,574 KB) [v2] Sat, 22 Aug 2026 19:47:37 UTC (3,575 KB) Full-text links: Access Paper: View PDF TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-06 Change to browse by: math math.NT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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 (opens in new tab) (opens in new tab) (opens in new tab)

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