Explicit algebraic solution of Zolotarev's First Problem for low-degree polynomials

Abstract

E.I. Zolotarev's classical so-called First Problem (ZFP), which was posed to him by P.L. Chebyshev, is to determine, for a given \(n\in{\mathbb N}\backslash\{1\}\) and for a given \(s\in{\mathbb R}\backslash\{0\}\), the monic polynomial solution \(Z^{*}_{n,s}\) to the following best approximation problem: Find\[\min_{a_k}\max_{x\in[-1,1]}|a_0+a_1 x+\dots+a_{n-2}x^{n-2}+(-n s)x^{n-1}+x^n|,\]where the \(a_k, 0\le k\le n-2\), vary in \(\mathbb R\). It suffices to consider the cases \(s>\tan^2\left(\pi/(2n)\right)\). In 1868 Zolotarev provided a transcendental solution for all \(n\geq2\) in terms of elliptic functions. An explicit algebraic solution  in power form to ZFP, as is suggested by the problem statement, is available only for \(2\le n\le 5.^1\) We have now obtained an explicit algebraic solution to ZFP for \(6\le n\le 12\) in terms of roots of dedicated polynomials. In this paper, we provide our findings for \(6\le n\le 7\) in two alternative fashions, accompanied by concrete examples. The cases \(8\le n\le 12\) we treat, due to their bulkiness, in a separate web repository. \(^1\) Added in proof: But see our recent one-parameter power form solution for \(n=6\) in  [38].