Characterizing meromorphic pseudo-lemniscates

TL;DR

This paper establishes criteria and tests for identifying pseudo-lemniscates of meromorphic functions based on preimage counting, aiding in understanding the geometric structure of these functions' level sets.

Contribution

It introduces new criteria and a testing method for recognizing pseudo-lemniscates of meromorphic functions using preimage multiplicities.

Findings

Criteria for a curve to be a pseudo-lemniscate based on preimage counts.

A test to determine if the image of a Jordan curve under a meromorphic function is not a Jordan curve.

Characterization of pseudo-lemniscates in simply connected domains.

Abstract

Let $f$ be a meromorphic function with simply connected domain $G\subset\mathbb{C}$, and let $\Gamma\subset\mathbb{C}$ be a smooth Jordan curve. We call a component of $f^{-1}(\Gamma)$ in $G$ a $\Gamma$-$pseudo$-$lemniscate$ of $f$. In this note we give criteria for a smooth Jordan curve $\mathcal{S}$ in $G$ (with bounded face $D$) to be a psuedo-lemniscate of $f$ in terms of the number of preimages (counted with multiplicity) which a given $w$ has under $f$ in $D$, as $w$ ranges over the Riemann sphere. We also develop a test, in the same terms, by which one may show that the image of a Jordan curve under $f$ is not a Jordan curve.