An explicit univariate and radical parametrization of the septic proper Zolotarev polynomials in power form

Abstract

The problem of determining an explicit one-parameter power form representation of the proper $n$-th degree Zolotarev polynomials 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. The next higher degrees, $n=5$ and $n=6$, were resolved only recently, by G. Grasegger and N. Th. Vo (2017) respectively by the present authors (2019). In this paper we settle the case $n=7$ using symbolic computation. The parametrization for the degrees $n\in \{2,3,4\}$ is a rational one, whereas for $n\in \{5,6,7\}$ it is a radical one. However, the case $n=7$ among the radical parametrizations requires special attention, since it is not a simple radical one.