The second Zolotarev case in the Erdös-Szegö solution to a Markov-type extremal problem of Schur

Schur's [20] Markov-type extremal problem is to determine (i) $M_n= \sup_{-1\leq \xi\leq 1}\sup_{P_n\in\mathbf{B}_{n,\xi,2}}(|P_n^{(1)}(\xi)| / n^2)$, where $\mathbf{B}_{n,\xi,2}=\{P_n\in\mathbf{B}_n:P_n^{(2)}(\xi)=0\}\subset \mathbf{B}_n=\{P_n:|P_n(x)|\leq 1 \;\textrm{for}\; |x| \leq 1\}$ and $P_n$ is an algebraic polynomial of degree $\leq n$. Erdos and Szego [4] found that for $n\geq 4$ this maximum is attained if $\xi=\pm 1$ and $P_n\in\mathbf{B}_{n,\pm 1,2}$ is a (unspecified) member of the one-parameter family of hard-core Zolotarev polynomials. An extremal such polynomial as well as the constant $M_n$ we have explicitly specified for $n=4$ in [17], and in this paper we strive to obtain an analogous amendment to the Erdos-Szego solution for $n = 5$. The cases $n>5$ still remain arcane. Our approach is based on the quite recently discovered explicit algebraic power form representation [6], [7] of the quintic hard-core Zolotarev polynomial, $Z_{5,t}$, to which we add here explicit descriptions of its critical points, the explicit form of Pell's (aka: Abel's) equation, as well as an alternative proof for the range of the parameter, $t$. The optimal $t=t^*$ which yields $M_5 = |Z_{5,t^*}^{(1)}(1)|/25$ we identify as the negative zero with smallest modulus of a minimal $P_{10}$. We then turn to an extension of (i), to higher derivatives as proposed by Shadrin [22], and we provide an analogous solution for $n=5$. Finally, we describe, again for $n = 5$, two new algebraic approaches towards a solution to Zolotarev's so-called first problem [2], [24] which was originally solved by means of elliptic functions

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

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]

On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation

ACM Computing Classification System (1998): G.1.1, G.1.2.We consider optimal Lagrange interpolation with polynomials of degree at most two on the unit interval [−1, 1]. In a largely unknown paper, Schurer (1974, Stud. Sci. Math. Hung. 9, 77-79) has analytically described the infinitely many zero-symmetric and zero-asymmetric extremal node systems −1 ≤ x1 < x2 < x3 ≤ 1 which all lead to the minimal Lebesgue constant 1.25 that had already been determined by Bernstein (1931, Izv. Akad. Nauk SSSR 7, 1025-1050). As Schurer’s proof is not given in full detail, we formally verify it by providing two new and sound proofs of his theorem with the aid of symbolic computation using quantifier elimination. Additionally, we provide an alternative, but equivalent, parameterized description of the extremal node systems for quadratic Lagrange interpolation which seems to be novel. It is our purpose to bring the computer-assisted solution of the first nontrivial case of optimal Lagrange interpolation to wider attention and to stimulate research of the higher-degree cases. This is why our style of writing is expository

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

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]

Confirmation of the topology of the Wendelstein 7-X magnetic field to better than 1:100,000

Fusion energy research has in the past 40 years focused primarily on the tokamak concept, but recent advances in plasma theory and computational power have led to renewed interest in stellarators. The largest and most sophisticated stellarator in the world, Wendelstein 7-X (W7-X), has just started operation, with the aim to show that the earlier weaknesses of this concept have been addressed successfully, and that the intrinsic advantages of the concept persist, also at plasma parameters approaching those of a future fusion power plant. Here we show the first physics results, obtained before plasma operation: that the carefully tailored topology of nested magnetic surfaces needed for good confinement is realized, and that the measured deviations are smaller than one part in 100,000. This is a significant step forward in stellarator research, since it shows that the complicated and delicate magnetic topology can be created and verified with the required accuracy