How many Zolotarev fractions are there?

TL;DR

This paper explores the classification of Zolotarev fractions based on their critical points and values, extending properties known for Chebyshev polynomials to a broader class of rational functions.

Contribution

It identifies and lists classes of Zolotarev fractions characterized by simple critical points and four critical values, expanding understanding of their structure.

Findings

Zolotarev fractions have multiple classes distinguished by critical points and values

The classification parallels Chebyshev polynomials but with more complexity

Many classes of Zolotarev fractions are enumerated and described

Abstract

Known properties of Chebyshev polynomials are the following: they have simple critical points with only two (finite) critical values. Those properties uniquely determine the named polynomials modulo affine transformations of dependent and independent variables. A similar property of Zolotarev fractions: simple critical points and only four critical values generates already many classes of rational functions modulo projective transformations of their dependent and independent variables. They are listed in this note.