Notes on the boundaries of quadrature domains

TL;DR

This paper explores the intrinsic link between classical quadrature domains and removable singularities of analytic sets in several complex variables, offering new insights into their boundary properties and singularities.

Contribution

It introduces a novel framework connecting quadrature domains with analytic set singularities, enhancing understanding of boundary algebraicity and singularity types.

Findings

Quadrature domain boundaries are algebraic.

The defining polynomial of a quadrature domain can be better characterized.

Boundary singularities of quadrature domains are classified within this new framework.

Abstract

We highlight an intrinsic connection between classical quadrature domains and the well-studied theme of removable singularities of analytic sets in several complex variables. Exploiting this connection provides a new framework to recover several basic properties of such domains, namely the algebraicity of their boundary, a better understanding of the associated defining polynomial and the possible boundary singularities that can occur.