Toric vector bundles and the number of zeros of vertical systems

[2608.27019] Toric vector bundles and the number of zeros of vertical systems Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2608.27019 (math) [Submitted on 27 Aug 2026 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Toric vector bundles and the number of zeros of vertical systems Authors: Carles Checa View PDF HTML (experimental) Abstract: We provide necessary and sufficient conditions for the specializations of a vertically parametrized polynomial system to attain the maximal number of complex nonzero solutions. To each vertical system, we attach a pair consisting of a projective simplicial toric variety and a toric vector bundle. The toric vector bundle allows for a homogenization of the polynomials in the Cox ring of the toric variety, providing a homogeneous ideal with the same zeros over the torus as the original system. We show that the maximal number of isolated solutions is attained if and only if this ideal has no solutions in the faces of the toric variety and prove that this happens for generic values of the parameters. In addition, we provide a novel formula for this generic number of zeros as an alternating sum of mixed volumes over a family of polytopes attached to the toric vector bundle. Comments: Comments welcome ;) Subjects: Algebraic Geometry (math.AG) Cite as: arXiv:2608.27019 [math.AG] (or arXiv:2608.27019v2 [math.AG] for this version) https://doi.org/10.48550/arXiv.2608.27019 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Carles Checa Mr. [ view email ] [v1] Thu, 27 Aug 2026 12:04:50 UTC (33 KB) [v2] Fri, 28 Aug 2026 07:31:13 UTC (34 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2026-08 Change to browse by: 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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