Adjoints and Canonical Forms of Polypols

TL;DR

This work develops a cohesive algebro-geometric framework for polypols, real domains with nonlinear boundary hypersurfaces, by proving the existence and uniqueness of adjoint curves $A_P$ and canonical forms $\Omega(P)$ for planar polypols (notably quasi-regular ones) and connecting them to positive geometries. It provides explicit formulas, notably $\Omega(P)=\frac{\alpha_P}{f_1\cdots f_k}\,dx\wedge dy$, and establishes a degree relation $\deg A_P=\deg(\partial P)-3$, while detailing the real topology of adjoints for convex polygons and many configurations of three ellipses. The paper then characterizes when the adjoint map is finite across degree patterns, computes adjoint-degree phenomena for heptagons via numerical monodromy (conjecturing a general 864-fold correspondence with general quartics), and links these geometric constructions to algebraic statistics through likelihood equations and push-forward of canonical forms. Extending to 3D, the authors formulate criteria for 3D polypols with quadric boundaries, and prove existence/uniqueness of adjoints for simple quadric polyhedral polypols, paralleling the planar theory. The outlook outlines key open problems, including Wachspress’s conjecture in broader settings, hyperbolicity criteria, singular adjoints, adjoint maps in more general types, and higher-dimensional generalizations, highlighting the foundational role of adjoints and canonical forms in the geometry of polypols and their statistical interpretations.

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.

Figures (31)

• Figure 1: Three polypols (shaded in orange) and their adjoints (purple). The boundary curves and the vertices of the polypols are shown in black. The singular points and "outside" intersection points of the boundary curves are purple.
• Figure 2: The ampersand curve given by $(y^2- x^2)(x - 1)(2x - 3) - 4(x^2 + y^2 - 2x)^2 = 0$ bounds a semi-algebraic set with a unique adjoint line.
• Figure 3: Polypol defined by three conics and vertices $A$, $B$, $C$. The adjoint curve is the union of three lines.
• Figure 4: A positive geometry bounded by a parabola and a cubic with a cusp which is a vertex. The canonical form is equal to $\frac{x^2+xy+y^2+x}{(y-x^2)(x^2-y^3)}\,\,dx\wedge dy$. Note that the adjoint curve is tangent to the cusp.
• Figure 5: Positive geometry with 2 boundary curves and 4 vertices.

Theorems & Definitions (96)

• Definition 2.1
• Proposition 2.2
• proof
• Example 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Example 2.8
• Definition 2.9