A $1$-point Quadrature domain of order $1$ not biholomorphic to a balanced domain

TL;DR

This paper constructs a specific 1-point Quadrature domain of order 1 that cannot be transformed into a balanced domain via biholomorphic maps, challenging Bell's conjecture about their origins.

Contribution

It provides a counterexample to Bell's conjecture by explicitly constructing a 1-point Quadrature domain of order 1 not biholomorphic to any balanced domain.

Findings

Counterexample to Bell's conjecture

Existence of non-biholomorphic 1-point Quadrature domains

Insights into the structure of Quadrature domains

Abstract

It is known that if $f: D_1 \to D_2$ is a polynomial biholomorphism with polynomial inverse and constant Jacobian then $D_1$ is a $1$-point Quadrature domain (the Bergman span contains all holomorphic polynomials) of order $1$ whenever $D_2$ is a balanced domain. Bell conjectured that all $1$-point Quadrature domains arise in this manner. In this note, we construct a $1$-point Quadrature domain of order $1$ that is not biholomorphic to any balanced domain.