Adjoints and Canonical Forms of Polypols
ABSTRACT: 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.
The code included here illustrates some results from Section 4 in the paper, in which we investigate types of polypols with a finite adjoint map. Of all types illustrated in the figure below, we show that (1,3,1,3) is the only one that does not have this property, and the other 7 types are the only ones that do (Theorem 4.2 and Proposition 4.5).
Project page created: 25/08/2021
Project contributors: Kathlén Kohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Rainer Sinn, Miruna-Stefana Sorea, Simon Telen
Corresponding author of this page: Simon Telen, Simon.Telen@mis.mpg.de
Software used: Julia (Version 1.5.2)