[2311.11113] Isotopy classification of Morse polynomials of degree 4 in ${\mathbb R}^2$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Algebraic Geometry arXiv:2311.11113 (math) [Submitted on 18 Nov 2023 ( v1 ), last revised 15 Jul 2026 (this version, v12)] Title: Isotopy classification of Morse polynomials of degree 4 in ${\mathbb R}^2$ Authors: V.A. Vassiliev View a PDF of the paper titled Isotopy classification of Morse polynomials of degree 4 in ${\mathbb R}^2$, by V.A. Vassiliev View PDF HTML (experimental) Abstract: We introduce a system of invariants of isotopy classes of Morse polynomials ${\mathbb R}^2 \to {\mathbb R}^1$, prove its completeness for polynomials of degrees $\leq 4$, calculate all 71 possible values of these invariants for the case of degree 4, and realize them by concrete Morse polynomials. Also we count all 45460 classes of {\em strictly} Morse polynomials of degree four with the maximal possible number (nine) of real critical points. Keywords: real algebraic geometry, Morse function, Milnor fiber, Coxeter-Dynkin graph, vanishing cycle, topological invariant, surgery, Lyashko--Looijenga map. Subjects: Algebraic Geometry (math.AG) MSC classes: 14P99, 14Q30, 14B07, 32S15 Cite as: arXiv:2311.11113 [math.AG] (or arXiv:2311.11113v12 [math.AG] for this version) https://doi.org/10.48550/arXiv.2311.11113 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Victor Vassiliev [ view email ] [v1] Sat, 18 Nov 2023 16:25:13 UTC (26 KB) [v2] Sun, 3 Dec 2023 09:01:07 UTC (26 KB) [v3] Sun, 19 May 2024 10:02:12 UTC (31 KB) [v4] Wed, 12 Jun 2024 14:10:58 UTC (32 KB) [v5] Thu, 29 Aug 2024 09:54:58 UTC (33 KB) [v6] Thu, 3 Oct 2024 13:44:27 UTC (45 KB) [v7] Wed, 9 Oct 2024 13:57:18 UTC (47 KB) [v8] Thu, 17 Apr 2025 16:45:58 UTC (49 KB) [v9] Wed, 14 May 2025 06:25:38 UTC (49 KB) [v10] Fri, 21 Nov 2025 14:08:33 UTC (49 KB) [v11] Fri, 24 Apr 2026 08:16:34 UTC (50 KB) [v12] Wed, 15 Jul 2026 10:08:40 UTC (51 KB) Full-text links: Access Paper: View a PDF of the paper titled Isotopy classification of Morse polynomials of degree 4 in ${\mathbb R}^2$, by V.A. Vassiliev View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2023-11 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