On the exponential transform of lemniscates

TL;DR

This paper demonstrates that the exponential transform of lemniscate domains is generally not rational, contrasting with quadrature domains, and employs polynomial and meromorphic resultants as key tools.

Contribution

It establishes that lemniscate domains do not have rational exponential transforms, expanding understanding of their complex analysis properties.

Findings

Exponential transforms of lemniscate domains are not rational functions.

Quadrature domains have rational exponential transforms, unlike lemniscate domains.

Polynomial and meromorphic resultants are used to analyze these transforms.

Abstract

It is known that the exponential transform of a quadrature domain is a rational function for which the denominator has a certain separable form. In the present paper we show that the exponential transform of lemniscate domains in general are not rational functions, of any form. Several examples are given to illustrate the general picture. The main tool used is that of polynomial and meromorphic resultants.