# An explicit univariate and radical parametrization of the sextic proper Zolotarev polynomials

## TL;DR

This paper derives an explicit univariate and radical parametrization for the sextic proper Zolotarev polynomials, extending the known explicit forms for degrees 2, 3, 5, and now 6, which are important in approximation theory.

## Contribution

It provides the first explicit radical parametrization for degree 6 proper Zolotarev polynomials, completing the known explicit forms for degrees 2, 3, and 5.

## Key findings

- Explicit radical parametrization for degree 6 Zolotarev polynomials
- Extension of known explicit forms for degrees 2, 3, and 5
- Advancement in the explicit representation of extremal polynomials

## Abstract

The problem to determine an explicit one-parameter power form representation of the proper Zolotarev polynomials of degree $n$ and with uniform norm $1$ on $[-1,1]$ can be traced back to P. L. Chebyshev. It turned out to be complicated, even for small values of $n$. Such a representation was known to A. A. Markov (1889) for $n=2$ and $n=3$. But already for $n=4$ it seems that nobody really believed that an explicit form can be found. As a matter of fact it was, by V. A. Markov in 1892, as A. Shadrin put it in 2004. About 125 years passed before an explicit form for the next higher degree, $n=5$, was found, by G. Grasegger and N. Th. Vo (2017). In this paper we settle the case $n=6$.