Conformal models and fingerprints of pseudo-lemniscates

TL;DR

This paper characterizes conformal equivalence of meromorphic functions on Jordan domains to rational maps with minimal degree and explores fingerprints of polynomial pseudo-lemniscates, highlighting boundary conditions' importance.

Contribution

It establishes conditions under which meromorphic functions are conformally equivalent to minimal degree rational maps and generalizes fingerprint characterization of polynomial pseudo-lemniscates.

Findings

Meromorphic functions on Jordan domains can be conformally equivalent to minimal degree rational maps under boundary conditions.

Minimal degree equivalence fails without boundary assumptions.

Fingerprint characterization of polynomial pseudo-lemniscates is extended.

Abstract

We prove that every function that is meromorphic on the closure of an analytic Jordan domain and sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the minimality of the degree fails in general without the boundary assumptions. As an application, we generalize a theorem of Ebenfelt, Khavinson and Shapiro by characterizing fingerprints of polynomial pseudo-lemniscates.