Computing singular solutions of polynomial systems: towards superlinear convergence without deflation

[2607.06329] A superlinear arclength endgame for corank-one singular polynomial systems Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2607.06329 (math) [Submitted on 7 Jul 2026 ( v1 ), last revised 2 Oct 2026 (this version, v4)] Title: A superlinear arclength endgame for corank-one singular polynomial systems Authors: Mikhail Karapetyants , Vladimir Kolmogorov , Jeferson Zapata View PDF HTML (experimental) Abstract: In Numerical Algebraic Geometry (NAG) isolated solutions of polynomial systems are usually computed by tracking a solution curve defined by a homotopy equation. The tracking problem becomes especially challenging close to a singular root (the ``endgame'' regime). Existing approaches include power series endgames, Cauchy endgames, and various methods that regularize the system via dual-space-based {\em deflation}. We introduce a new ``Arclength Endgame'' for corank-1 systems which combines the idea of the classical {\em pseudo-arclength continuation method} with the estimation of the Puiseux series of the curve. We formally prove that it has an R-superlinear rate of convergence in some neighborhood of the root. The method uses only evaluations of the system and its Jacobian. In contrast, previous homotopy techniques with proven superlinear convergence (such as deflation) require computing additional derivatives of the system, or need additional nondegeneracy assumptions involving higher-order derivatives of the system at the singular root. A key step in our approach (as well as in the standard power series endgame) is estimating the ratio $k_1/c$ of the Puiseux series of the curve. We present a new method for that, and discuss its advantages over previous methods. Subjects: Numerical Analysis (math.NA) Cite as: arXiv:2607.06329 [math.NA] (or arXiv:2607.06329v4 [math.NA] for this version) https://doi.org/10.48550/arXiv.2607.06329 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Vladimir Kolmogorov [ view email ] [v1] Tue, 7 Jul 2026 14:26:11 UTC (1,575 KB) [v2] Fri, 28 Aug 2026 11:43:29 UTC (1,579 KB) [v3] Tue, 22 Sep 2026 13:58:06 UTC (1,560 KB) [v4] Fri, 2 Oct 2026 14:08:39 UTC (329 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-07 Change to browse by: cs cs.NA math 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)

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