Adjoints and canonical forms of polypols
Kathlén Kohn
KTH Royal Institute of Technology, Stockholm, SwedenRagni Piene
University of Oslo, Oslo, NorwayKristian Ranestad
University of Oslo, Oslo, NorwayFelix Rydell
KTH Royal Institute of Technology, Stockholm, SwedenBoris Shapiro
Stockholm University, Stockholm, SwedenRainer Sinn
Universität Leipzig, Leipzig, GermanyMiruna-Ştefana Sorea
Lucian Blaga University of Sibiu, Sibiu, RomaniaSimon Telen
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Abstract
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.
Cite this article
Kathlén Kohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Rainer Sinn, Miruna-Ştefana Sorea, Simon Telen, Adjoints and canonical forms of polypols. Doc. Math. 30 (2025), no. 2, pp. 275–346
DOI 10.4171/DM/991