Geometry of Adjoint Hypersurfaces for Polytopes
Clemens Brüser, Julian Weigert
TL;DR
This work defines the adjoint hypersurface of a convex polytope without relying on simple hyperplane arrangements by encoding vanishing data through orders along linear spaces ${\mathcal L}(P)$ and points in the point residual ${\mathcal R}_0(P)$. It proves the adjoint is, up to scaling, the unique homogeneous polynomial of degree $d-n-1$ vanishing along every $L\in {\mathcal L}(P)$ with order at least ${\operatorname{ord}}_P(L)$, and further shows a reduction to interpolation at zero-dimensional data. The existence is established via triangulations of the dual polytope and careful factor-counting, while uniqueness leverages the canonical form in the theory of positive geometries. These results generalize prior simple-arrangement theorems to a broad class of polytopes and illuminate singularities of the adjoint through residual geometry, with practical computational demonstrations for explicit polytopes.
Abstract
In this article we prove that the adjoint polynomial of arbitrary convex polytopes is up to scaling uniquely determined by vanishing to the right order on the polytopes residual arrangement. This answers a problem posed by Kohn and Ranestad and generalizes their main theorem to non-simple polytopes. We furthermore prove that the adjoint polynomial is already characterized by vanishing to the right order on a zero-dimensional subset of the residual arrangement.