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

Abstract

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.