Zolotarev polynomials of degree 5, 6 and 7 with simple critical points and their moduli spaces

TL;DR

This paper investigates Zolotarev polynomials of degrees 5, 6, and 7 with simple critical points, defining combinatorial moduli spaces that mirror analytical properties but are easier to construct.

Contribution

It introduces and studies combinatorial moduli spaces for Zolotarev polynomials of specific degrees, providing a new approach to understanding their structure.

Findings

Defined combinatorial moduli spaces for degrees 5, 6, and 7

Showed these spaces share properties with analytical moduli spaces

Simplified the construction of moduli spaces for these polynomials

Abstract

A polynomial $p\in \mathbb{C}[z]$ with three finite values is called the Zolotarev polynomial. For a class of such polynomials with the given degree, given passport and simple critical points we define a \emph{combinatorial moduli space}. A combinatorial moduli space have the same essential properties as analytical moduli space, but much easier to construct. We study these objects for Zolotarev polynomials of degree 5, 6 and 7.