The Geometry of Polycons and a Counterexample to Wachspress' Conjecture

1. Problem Statement and Context

Background:
Polycons are a generalization of polygons introduced by Eugene Wachspress (1975) for use in finite element methods. Unlike standard polygons, polycons allow boundary components to be segments of conic sections (degree 1 or 2 curves). Associated with every polycon $P$ is a unique adjoint curve $A$, defined as the curve of minimal degree that vanishes on the residual arrangement $R(P)$ (the set of singular points of the algebraic boundary that are not vertices).

The Conjecture:
Wachspress conjectured that for a "regular" real polycon, the real part of its adjoint curve $A(\mathbb{R})$ does not intersect the interior of the polycon ($P_{>0}$). This property is crucial for the stability and positivity of basis functions in finite element analysis.

• Status: The conjecture was proven true for convex polygons. However, for polycons bounded by three conics (degree 6 polycons), the conjecture remained open, with only partial results suggesting it might hold in most cases.

The Core Question:
Does there exist a regular real polycon bounded by three conics whose adjoint curve has a real oval lying entirely within the interior of the polycon?

2. Methodology

The author employs a blend of algebraic geometry, specifically the theory of determinantal representations and divisor theory, to construct a counterexample and analyze the geometry of adjoint maps.

Key Methodological Steps:

1. Construction of a Counterexample:

   • The author explicitly constructs a specific polycon $P$ bounded by three conics ($C_1, C_2, C_3$) with specific vertices and coefficients.
   • Using computer algebra (Macaulay2), the author verifies that $P$ is a regular nodal polycon (its boundary has only node singularities and no residual points on the Euclidean boundary).
   • The adjoint polynomial $\alpha(x,y)$ is computed. The author demonstrates that $\alpha$ changes sign within the interior of $P$, proving the existence of a zero (and thus an oval) inside the polycon.

2. Theoretical Framework: Determinantal Representations:

   • The paper establishes a deep link between polycons and symmetric linear determinantal representations (LDRs).
   • Contact Curves: The author proves that if a conic boundary component of a polycon $P$ is replaced by a line, the adjoint curve $A'$ of the new polycon is a contact curve to the original adjoint $A$. This means their intersection divisor is even ($A' \cdot A = 2D$).
   • Dixon's Algorithm: The paper utilizes Dixon's algorithm to construct symmetric LDRs from contact divisors. This allows the adjoint curve to be represented as the determinant of a symmetric matrix of linear forms ($M$).

3. Analysis of Adjoint Maps:

   • The author investigates the adjoint map $\alpha_{2,2,2}$, which maps a polycon bounded by three conics to its cubic adjoint.
   • By relating the fibers of this map to the space of symmetric LDRs (modulo scaling and change of basis), the author characterizes the families of polycons that share the same adjoint.

3. Key Contributions and Results

A. Counterexample to Wachspress' Conjecture (Theorem 2.11)

The primary result is the disproof of Wachspress' Conjecture.

• The Example: A specific polycon bounded by three ellipses with vertices $v_{12}=(9,-6)$, $v_{23}=(0,0)$, and $v_{31}=(-3,-6)$.
• The Finding: The adjoint curve of this polycon contains an oval (a closed loop) that lies strictly within the interior of the polycon.
• Significance: This demonstrates that the "positivity" property of adjoints does not hold universally for all regular polycons, specifically those bounded by three conics.

B. Geometry of Adjoint Maps and Determinantal Representations

• Theorem 4.3: Proves that the adjoint $A'$ of a polycon $P'$ (obtained by replacing a conic boundary of $P$ with a line) is a contact curve to the original adjoint $A$. Specifically, $A' \cdot A = 2(R(P) \cap R(P'))$.
• Proposition 4.8 & Corollary 4.10: Establishes a bijection between the fibers of the adjoint map $\alpha_{2,2,2}$ and the equivalence classes of symmetric linear determinantal representations of the cubic adjoint curve.
  • This implies that the variety $Y_{2,2,2}$ (the space of polycons bounded by three conics) is irreducible.
  • It provides an explicit parametrization of all polycons sharing the same adjoint curve via the action of $GL_3(k)$ on the determinantal matrix.

C. Generic Smoothness (Corollary 4.14)

• The paper proves that generically, the adjoint of a polycon bounded by three conics is a smooth cubic curve.
• This is derived from the fact that the image of the adjoint map contains all smooth cubic curves (which admit symmetric LDRs), and the domain variety is irreducible.

D. Refined Conjecture (Conjecture 2.12)

• While the general conjecture is false, the author proposes a refined version: Wachspress' Conjecture might still hold for polycons where replacing any boundary conic with a line results in a regular polycon. The counterexample provided fails this specific regularity condition.

4. Significance

1. Resolution of an Open Problem: The paper definitively closes the question of Wachspress' Conjecture for polycons bounded by three conics, showing it is false in the general case.
2. Bridging Fields: It successfully bridges the gap between applied mathematics (finite element methods/polycons) and pure algebraic geometry (determinantal representations, divisor classes, and moduli spaces).
3. New Geometric Tools: The identification of the relationship between "conic-to-line" replacements and contact curves provides a powerful new tool for analyzing adjoint curves. It allows researchers to use the rich theory of symmetric matrices to study the topology of polycon adjoints.
4. Implications for FEM: For finite element methods, this result suggests that basis functions derived from polycon adjoints may not always be positive in the interior for complex geometries, necessitating caution or alternative constructions in specific applications.

Summary

Clemens Brüser's paper uses the machinery of symmetric linear determinantal representations to construct a specific counterexample disproving Wachspress' Conjecture regarding the non-vanishing of adjoint curves in the interior of polycons. Beyond the counterexample, the work provides a comprehensive geometric description of the fibers of the adjoint map for polycons bounded by three conics, proving the irreducibility of the underlying parameter space and establishing that generic adjoints are smooth.