Shapes, fingerprints and rational lemniscates

TL;DR

This paper provides a new proof linking polynomial and rational lemniscates to their fingerprints, showing that polynomial lemniscates' fingerprints are roots of Blaschke products, and extends this to rational lemniscates.

Contribution

It offers a simplified proof of a known theorem and generalizes it from polynomial to rational lemniscates, addressing an open problem.

Findings

Fingerprints of polynomial lemniscates are roots of Blaschke products.

The proof extends to rational lemniscates.

Establishes a correspondence between lemniscates and their fingerprints.

Abstract

It has been known since the work of A.A. Kirillov that any smooth Jordan curve in the plane can be represented by its so-called fingerprint, an orientation preserving smooth diffeomorphism of the unit circle onto itself. In this paper, we give a new, simple proof of a theorem of Ebenfelt, Khavinson and Shapiro stating that the fingerprint of a polynomial lemniscate of degree $n$ is given by the $n$-th root of a Blaschke product of degree $n$ and that conversely, any smooth diffeomorphism induced by such a map is the fingerprint of a polynomial lemniscate of the same degree. The proof is easily generalized to the case of rational lemniscates, thus solving a problem raised by the previously mentioned authors.