Geodesic
Adjoints and canonical forms of polypols
Kathlén Kohn1 orcid ; Ragni Piene2 orcid ; Kristian Ranestad2 orcid ; Felix Rydell1 orcid ; Boris Shapiro3 orcid ; Rainer Sinn4 orcid ; Miruna-Ştefana Sorea5 orcid ; Simon Telen6 orcid
1KTH Royal Institute of Technology, Stockholm, Sweden
2University of Oslo, Oslo, Norway
3Stockholm University, Stockholm, Sweden
4Universität Leipzig, Leipzig, Germany
5Lucian Blaga University of Sibiu, Sibiu, Romania
6Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Documenta mathematica, Tome 30 (2025) no. 2, pp. 275-346

Voir la notice de l'article provenant de la source EMS Press

Polypols are natural generalizations of polytopes, with boundaries given by non-linear 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.
DOI : 10.4171/dm/991
Classification : 14N30, 14H10, 14P25, 14H05, 14H50, 14P10, 14H81, 51M20
Mots-clés : polypols, adjoints, canonical forms, positive geometries, plane curves, algebraic statistics

Kathlén Kohn  1   ; Ragni Piene  2   ; Kristian Ranestad  2   ; Felix Rydell  1   ; Boris Shapiro  3   ; Rainer Sinn  4   ; Miruna-Ştefana Sorea  5   ; Simon Telen  6

1 KTH Royal Institute of Technology, Stockholm, Sweden
2 University of Oslo, Oslo, Norway
3 Stockholm University, Stockholm, Sweden
4 Universität Leipzig, Leipzig, Germany
5 Lucian Blaga University of Sibiu, Sibiu, Romania
6 Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany 
Kathlén Kohn; Ragni Piene; Kristian Ranestad; Felix Rydell; Boris Shapiro; Rainer Sinn; Miruna-Ştefana Sorea; Simon Telen. Adjoints and canonical forms of polypols. Documenta mathematica, Tome 30 (2025) no. 2, pp. 275-346. doi: 10.4171/dm/991
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