Adjoints and Canonical Forms of Polypols
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arXiv
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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| Soggetti: | |
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| _version_ | 1866908275543375872 |
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| author | Kohn, Kathlén Piene, Ragni Ranestad, Kristian Rydell, Felix Shapiro, Boris Sinn, Rainer Sorea, Miruna-Stefana Telen, Simon |
| author_facet | Kohn, Kathlén Piene, Ragni Ranestad, Kristian Rydell, Felix Shapiro, Boris Sinn, Rainer Sorea, Miruna-Stefana Telen, Simon |
| contents | Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_11747 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Adjoints and Canonical Forms of Polypols Kohn, Kathlén Piene, Ragni Ranestad, Kristian Rydell, Felix Shapiro, Boris Sinn, Rainer Sorea, Miruna-Stefana Telen, Simon Algebraic Geometry 14H10, 14N30, 14P25 (Primary) 14H05, 14H50, 14P10, 14H81, 51M20 (Secondary) Polypols are natural generalizations of polytopes, with boundaries given by nonlinear algebraic hypersurfaces. We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry. We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics. |
| title | Adjoints and Canonical Forms of Polypols |
| topic | Algebraic Geometry 14H10, 14N30, 14P25 (Primary) 14H05, 14H50, 14P10, 14H81, 51M20 (Secondary) |
| url | https://arxiv.org/abs/2108.11747 |