The Geometry of Polycons and a Counterexample to Wachspress' Conjecture

TL;DR

This paper explores the geometry of polycons, introduces a counterexample to Wachspress's conjecture regarding their adjoint curves, and analyzes the properties of these curves, especially in polycons bounded by three conics.

Contribution

It provides the first known counterexample to Wachspress's conjecture and offers a detailed geometric and algebraic analysis of polycons and their adjoint curves.

Findings

Counterexample polycon bounded by three conics disproves Wachspress's conjecture.

Explicit description of adjoint curves using symmetric linear determinantal representations.

Generically, the adjoint of a polycon bounded by three conics is smooth.

Abstract

Polycons, initially introduced by Wachspress in 1975 as a tool in finite element methods, are generalizations of polygons in that they allow conic boundary components. We are interested in the adjoint curve of a given polycon, i.e. the unique curve of minimal degree vanishing in the so-called residual arrangement. It was conjectured by Wachspress that under some regularity assumptions this curve does not vanish in the interior of its defining polycon. However, until recently the only class of polycons for which this was proven were convex polygons. We present a polycon bounded by three conics that constitutes a counterexample to Wachspress' conjecture. The origin of this counterexample reveals some beautiful geometry of polycons. Replacing one degree two boundary component of a polycon with a line produces a new polycon. We show that the adjoint of the latter is a contact curve to the adjoint of the former. This naturally leads to the consideration of symmetric linear determinantal representations of adjoints, which lets us explicitly describe the fibers of the adjoint map in the case of polycons bounded by three conics. As a corollary we prove that generically the adjoint of a polycon bounded by three conics is smooth.