Geodesic
Adjoints and canonical forms of polypols
Documenta mathematica, Tome 30 (2025) no. 2, pp. 275-346

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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 
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     title = {Adjoints and canonical forms of polypols},
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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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