[Paper Review] Adjoints and Canonical Forms of Polypols

Kathlén Kohn, Ragni Piene+6 more|arXiv (Cornell University)|Aug 26, 2021

Point processes and geometric inequalities4 citations

TL;DR

This paper establishes a foundational algebro-geometric framework for planar and 3D polypols—generalizations of polytopes with nonlinear algebraic boundaries—by studying their adjoint hypersurfaces and canonical differential forms. It proves that the adjoint curve of a convex polygon is a nonsingular cubic that does not intersect the polygon's interior, confirming a key case of Wachspress’s conjecture, and characterizes all planar polypols for which the adjoint map is finite, linking them to positive geometries and algebraic statistics.

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.

Motivation & Objective

• To establish polypols as a new class of geometric objects in complex and real algebraic geometry, generalizing polytopes with nonlinear algebraic boundaries.
• To investigate the existence, uniqueness, and real topology of adjoint hypersurfaces associated with polypols, particularly in the plane and 3-space.
• To connect the adjoint curve of a planar polypol to the numerator of the canonical differential form in positive geometry, as introduced in particle physics.
• To resolve cases of Wachspress’s conjecture on the non-intersection of the adjoint curve with the interior of regular planar polypols.
• To characterize all types of planar polypols for which the rational map sending a polypol to its adjoint is finite, and to explore connections with algebraic statistics and likelihood equations.

Proposed method

• Define rational polypols as bounded semialgebraic sets in R² or R³ with rational boundary curves, generalizing polytopes.
• Construct the adjoint curve as the minimal-degree curve passing through singular points and 'outside' intersection points of the boundary components.
• Relate the adjoint curve to the numerator of the canonical differential form in positive geometry via residue theory and global residue formulas.
• Use algebraic geometry tools such as residues, push-forwards, and trace tests to analyze canonical forms and likelihood equations.
• Apply computational algebraic geometry and symbolic computation (e.g., via Julia packages) to study configurations of conics and their adjoints.
• Employ topological and real algebraic geometry techniques to analyze the real topology of adjoint curves, particularly for convex polygons and three-conic configurations.

Experimental results

Research questions

• RQ1 Under what conditions does a planar polypol admit a unique adjoint curve, and when is the adjoint map finite?
• RQ2 Does the adjoint curve of a regular planar polypol fail to intersect its interior, as conjectured by Wachspress?
• RQ3 How can the canonical differential form of a positive geometry be expressed in terms of the adjoint hypersurface?
• RQ4 What is the real topology of the adjoint curve for convex polygons, and how does it vary with the number of sides?
• RQ5 Which real n-dimensional polypols admit a canonical differential form with poles on the boundary and zeros on an adjoint hypersurface?

Key findings

• The adjoint curve of a convex polygon is a nonsingular cubic curve that does not intersect the interior of the polygon, confirming a key case of Wachspress’s conjecture.
• For all convex polygons, the real topology of the adjoint curve is completely characterized: it consists of a single oval (a single connected component in the real projective plane).
• All planar polypols for which the adjoint map is finite are completely classified; these include specific configurations such as (1,3,1,3)-polypols and heptagons with certain boundary arrangements.
• The canonical differential form of a quasi-regular rational polypol is shown to be additive under decomposition into simpler polypols, supporting the structure of positive geometries.
• The push-forward of the canonical form under a morphism of positive geometries is shown to satisfy a trace test, and the conjecture on its global validity remains open.
• For three conics forming a polypol, the adjoint is a cubic curve, and its singularities arise precisely when the conics share a common point or are mutually tangent at a residual point in the 9-tuple of intersection points.