Canonical forms of polytopes from adjoints
Christian Gaetz
TL;DR
The work studies canonical forms of positive geometries arising from projectivized pointed polyhedral cones, showing that the numerator of the canonical form is the adjoint of the dual cone. By defining the adjoint $\mathrm{adj}_C$ via triangulations and proving its independence from the triangulation, the authors derive a facet-based expression and connect it to the canonical form $\Omega_P(x)=\frac{\mathrm{adj}_{P^{\vee}}(x)}{\prod_{F\in f(P)} (1-v_F \cdot x)}\,dx$, where $v_F$ are facet normals. They further interpret the adjoint as the minimal polynomial vanishing on the residual arrangement $\mathcal{R}_{C^{\vee}}$, with a dimension-by-dimension induction establishing vanishing on residuals. A uniqueness result is proved in the simple-hyperplane case, identifying $\mathrm{adj}_{C^{\vee}}$ as the defining equation of the corresponding hypersurface, while non-simple cases yield limits that may be non-reduced. Together, these results clarify how adjoints encode pole cancellation and provide a triangulation-free route to canonical forms for projective polytopes.
Abstract
Projectivizations of pointed polyhedral cones $C$ are positive geometries in the sense of Arkani-Hamed, Bai, and Lam. Their canonical forms look like $$ Ω_C(x)=\frac{A(x)}{B(x)} dx, $$ 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 $C$.