Non-algebraic quadrature domains

TL;DR

This paper demonstrates that in four dimensions, quadrature domains can have boundaries not defined by polynomials, challenging the classical understanding from two dimensions, using explicit constructions and elliptic integrals.

Contribution

It provides explicit four-dimensional examples of quadrature domains with non-algebraic boundaries, confirming a conjecture in higher dimensions.

Findings

Quadrature domains in 4D can have non-polynomial boundaries.

Explicit constructions use Schwarz potential and elliptic integrals.

Confirms conjecture about non-algebraic quadrature domains in higher dimensions.

Abstract

It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial. We show, by explicitly constructing some four-dimensional examples, that this is not always the case. This confirms, in dimension 4, a conjecture of the second author. Our method is based on the Schwarz potential and involves elliptic integrals of the third kind.