The Wachspress Counterexample

Wachspress conjectured in the 1970s that rational basis functions for polygonal finite elements generalize cleanly to regions bounded by algebraic curves — polycons. The conjecture has guided fifty years of work in geometric interpolation and computational geometry.

Brüser constructs a three-conic polycon that disproves it.

The counterexample is minimal: a region bounded by exactly three conics. Replacing one degree-two boundary component with a straight line produces a new polycon whose adjoint curve — the auxiliary curve that controls the basis function construction — becomes a contact curve to the original boundary. This geometric entanglement makes the Wachspress construction inconsistent.

The mechanism is degree-theoretic. Wachspress’s construction requires the adjoint curve to pass through certain intersection points with specific multiplicities. For polygons (line boundaries), the degree constraints are always satisfiable. For polycons with sufficiently many conic boundaries, the degrees overcount — there aren’t enough free parameters in the adjoint to satisfy all the constraints simultaneously.

The fifty-year gap between conjecture and counterexample reflects the difficulty: polycons with conic boundaries are rare in applications, and the failure only manifests at three or more conics. Polygons (the practical case) work perfectly. The generalization fails not because it was wrong-headed but because the algebraic complexity of conic intersections is qualitatively different from linear intersections.

The conjecture was true in the case that mattered. It was false in the generalization nobody needed — until someone tried it.