Geometry of Adjoint Hypersurfaces for Polytopes

TL;DR

This paper characterizes the adjoint polynomial of convex polytopes through its vanishing properties, extending previous results to non-simple polytopes and providing a unique determination method.

Contribution

It proves the uniqueness of the adjoint polynomial for arbitrary convex polytopes based on vanishing conditions, generalizing prior work to non-simple cases.

Findings

Adjoint polynomial is uniquely determined by vanishing order on the residual arrangement.

Extension of Kohn and Ranestad's theorem to non-simple polytopes.

Characterization of the adjoint polynomial via vanishing on a zero-dimensional subset.

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.