Chebyshev polynomials and navigation in the moduli space of hyperelliptic curves

P. L. Chebyshev posed a problem of finding a polynomial of least deviation on the closed see E of the real axis. He himself obtained the solution of this problem for E = [-1, 1]. The solution of the least deviation problem for E = [-1, a] boolean OR [b, 1], -1 < a < b < 1, followed from Zolotarev's works [14, 24, 25], and for many special cases it was obtained by Akhiezer [4]. The constructive solutions with the use of elliptic functions are given in these works. Some of the least deviation problems were solved by Lebedev [13] for the case in which E is a combination of many segments. Akhiezer proposed [3] to use in the general case Schottky automorphic functions for the parametric representation of the solutions. However, this approach (for extremal rational functions see [16]) has not so far yielded numerical results. In their recent work [19] Peherstorfer and Schiefermayr developed an alternative method of calculating polynomials of least deviation an many segments. In problems of computational mathematics polynomials of least deviation are used to optimize computational algorithms, in particular, iterative methods, quadrature formulae, and explicit difference schemes for ordinary differential equations [15].