Canonical forms of polytopes from adjoints

Abstract

Projectivizations of pointed polyhedral cones $C$ are positive geometries in the sense of Arkani-Hamed, Bai, and Lam. Their canonical forms look like $$\hat{I}{\text{c◯}}_C(x)=\frac{A(x)}{B(x)}dx,$$ with $A,B$ polynomials. The denominator $B(x)$ is just the product of the linear equations defining the facets of $C$. We will see that the numerator $A(x)$ is given by the adjoint polynomial of the dual cone $C^{\vee }$. The adjoint was originally defined by Warren, who used it to construct barycentric coordinates in general polytopes. Confirming the intuition that the job of the numerator is to cancel unwanted poles outside the polytope, we will see that the adjoint is the unique polynomial of minimal degree whose hypersurface contains the residual arrangement of non-face intersections of supporting hyperplanes of $C$.

These are lightly edited notes from a lecture given in February 2020, posted here by request, for ease of citation