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The Geometry of Polycons and a Counterexample to Wachspress' Conjecture

Clemens Brüser

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.

The Geometry of Polycons and a Counterexample to Wachspress' Conjecture

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.
Paper Structure (13 sections, 16 theorems, 38 equations, 5 figures)

This paper contains 13 sections, 16 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline of the Article
  3. Polycons, Adjoints, and Wachspress' Conjecture
  4. Polycons and their Adjoints
  5. Wachspress' Conjecture
  6. Adjoint Maps
  7. A Note on Positive Geometries
  8. Preliminaries on Determinantal Representations
  9. Results from Algebraic Geometry
  10. Contact Curves to the Adjoint
  11. Fibers of the Adjoint Map $\alpha_{2,2,2}$
  12. Code Accompanying \ref{['thm: WachspressWrong']}
  13. A Note on the Genesis of \ref{['ex: counterexample']}

Key Result

Proposition 2.6

For every polycon ${\mathcal{P}}$ there exists a unique adjoint curve $A_{\mathcal{P}}$. Furthermore $A_{\mathcal{P}}$ and the algebraic boundary $C_{\mathcal{P}}$ have no irreducible components in common.

Figures (5)

  • Figure 1: Examples of a regular (left) and an irregular (right) real polycon (shaded in orange). Both polycons are nodal.
  • Figure 2: A regular degree five polycon (orange) and its conic adjoint (purple).
  • Figure 3: Counterexample to Wachspress' Conjecture.
  • Figure 4: A regular degree six polycon (left), for which replacing one of the boundary conics with a line results in a degree five polycon that is not regular (right).
  • Figure 5: A polycon ${\mathcal{P}}$ (orange) with its adjoint (purple). The green conic is the adjoint of the polycon obtained by replacing the "inflated" side of ${\mathcal{P}}$ with the dashed line.

Theorems & Definitions (48)

  • Definition 2.1: Polycon
  • Definition 2.2: Nodal and regular polycons
  • Definition 2.3: Residual Arrangement
  • Definition 2.4: Adjoint
  • Remark 2.5
  • Proposition 2.6: KohnEtAl2024AdjCurves
  • Remark 2.7
  • Conjecture 2.8: Wachspress' Conjecture, Wachspress1980AdjConjecture, KohnEtAl2024AdjCurves
  • Lemma 2.9: KohnEtAl2024AdjCurves
  • Example 2.10: Counterexample to \ref{['conj: Wachspress']}
  • ...and 38 more