Geometry of Adjoint Hypersurfaces for Polytopes

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 [9, Problem 2] and generalizes their main Theorem [9, Theorem 1] 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.