[2306.02905] Inner and Partial non-degeneracy of mixed functions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Algebraic Geometry arXiv:2306.02905 (math) [Submitted on 5 Jun 2023] Title: Inner and Partial non-degeneracy of mixed functions Authors: Benjamin Bode , Eder L. Sanchez Quiceno View a PDF of the paper titled Inner and Partial non-degeneracy of mixed functions, by Benjamin Bode and Eder L. Sanchez Quiceno View PDF HTML (experimental) Abstract: Mixed polynomials $f:\mathbb{C}^2\to\mathbb{C}$ are polynomial maps in complex variables $u$ and $v$ as well as their complex conjugates $\bar{u}$ and $\bar{v}$. They are therefore identical to the set of real polynomial maps from $\mathbb{R}^4$ to $\mathbb{R}^2$. We generalize Mondal's notion of partial non-degeneracy from holomorphic polynomials to mixed polynomials, introducing the concepts of partially non-degenerate and strongly partially non-degenerate mixed functions. We prove that partial non-degeneracy implies the existence of a weakly isolated singularity, while strong partial non-degeneracy implies an isolated singularity. We also compare (strong) partial non-degeneracy with other types of non-degeneracy of mixed functions, such as (strong) inner non-degeneracy, and find that, in contrast to the holomorphic setting, the different properties are not equivalent for mixed polynomials. We then introduce additional conditions under which strong partial non-degeneracy becomes equivalent to the existence of an isolated singularity. Furthermore, we prove that mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition, resulting in an explicit Milnor (sphere) fibration. Comments: 35 pages, 4 figures Subjects: Algebraic Geometry (math.AG) ; Complex Variables (math.CV); Geometric Topology (math.GT) MSC classes: 14B05, 14J17, 14M25, 14P05, 32S05, 32S55 Cite as: arXiv:2306.02905 [math.AG] (or arXiv:2306.02905v1 [math.AG] for this version) https://doi.org/10.48550/arXiv.2306.02905 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Kodai Math. J. 48(3): 341-387 (October 2025) Related DOI : https://doi.org/10.2996/kmj48301 Focus to learn more DOI(s) linking to related resources Submission history From: Eder Leandro Sanchez Quiceno [ view email ] [v1] Mon, 5 Jun 2023 14:11:50 UTC (518 KB) Full-text links: Access Paper: View a PDF of the paper titled Inner and Partial non-degeneracy of mixed functions, by Benjamin Bode and Eder L. Sanchez Quiceno View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2023-06 Change to browse by: math math.CV math.GT 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