@article{Rack_Vajda_2019, title={Explicit algebraic solution of Zolotarev’s First Problem for low-degree polynomials}, volume={48}, url={https://ictp.acad.ro/jnaat/journal/article/view/1173}, DOI={10.33993/jnaat482-1173}, abstractNote={<p>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<br>\[<br>\min_{a_k}\max_{x\in[-1,1]}|a_0+a_1 x+\dots+a_{n-2}x^{n-2}+(-n s)x^{n-1}+x^n|,<br>\]<br>where the \(a_k, 0\le k\le n-2\), vary in \(\mathbb R\). It suffices to consider the cases \(s&gt;\tan^2\left(\pi/(2n)\right)\).</p> <p>In 1868 Zolotarev provided a transcendental solution for all \(n\geq2\) in terms of elliptic functions. An explicit algebraic solution&nbsp; in power form to ZFP, as is suggested by the problem statement, is available only for \(2\le n\le 5.^1\)</p> <p>We have now obtained an explicit algebraic solution to ZFP for \(6\le n\le 12\)&nbsp;in terms of roots of dedicated polynomials.</p> <p>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.</p> <p>\(^1\) <em>Added in proof:</em> But see our recent one-parameter power form solution for \(n=6\) in&nbsp; [38].</p>}, number={2}, journal={J. Numer. Anal. Approx. Theory}, author={Rack, Heinz Joachim and Vajda, Robert}, year={2019}, month={Dec.}, pages={175–201} }