Ewald's Conjecture and integer points in algebraic and symplectic toric geometry

[2310.10366] Ewald's Conjecture and integer points in algebraic and symplectic toric geometry Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2310.10366 (math) [Submitted on 16 Oct 2023 ( v1 ), last revised 20 Mar 2024 (this version, v3)] Title: Ewald's Conjecture and integer points in algebraic and symplectic toric geometry Authors: Luis Crespo , Álvaro Pelayo , Francisco Santos View a PDF of the paper titled Ewald's Conjecture and integer points in algebraic and symplectic toric geometry, by Luis Crespo and 2 other authors View PDF HTML (experimental) Abstract: We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the $2$-dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes. Comments: 37 pages, 6 figures. Added third co-author as a result of improvements. In particular we strenghten our previous theorem on Ewald's Conjecture by introducing deeply mononote polytopes and prove new cases of Nill's Conjecture Subjects: Combinatorics (math.CO) ; Algebraic Geometry (math.AG); Symplectic Geometry (math.SG) MSC classes: Primary 53D05, 53D20, 52A20, Secondary 52C07, 52B11, 52B20 Cite as: arXiv:2310.10366 [math.CO] (or arXiv:2310.10366v3 [math.CO] for this version) https://doi.org/10.48550/arXiv.2310.10366 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Proc. London Math. Soc., 132:4 (2026), e70144 Related DOI : https://doi.org/10.1112/plms.70144 Focus to learn more DOI(s) linking to related resources Submission history From: Luis Crespo [ view email ] [v1] Mon, 16 Oct 2023 13:04:49 UTC (250 KB) [v2] Thu, 26 Oct 2023 08:12:36 UTC (251 KB) [v3] Wed, 20 Mar 2024 16:28:15 UTC (256 KB) Full-text links: Access Paper: View a PDF of the paper titled Ewald's Conjecture and integer points in algebraic and symplectic toric geometry, by Luis Crespo and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2023-10 Change to browse by: math math.AG math.SG 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

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