Representation of moduli spaces of curves and calculation of extremal polynomials

The classical Chebyshev and Zolotarev polynomials are the first ranks of the hierarchy of extremal polynomials, which are typical solutions of problems on the conditional minimization of the uniform norm over a space of polynomials. In the general case such polynomials are connected with hyperelliptic curves the genus of which labels the ranks of the hierarchy. Representations of the moduli spaces of such curves are considered in this paper with applications to the calculation of extremal polynomials. Uniformizing curves by special Schottky groups one obtains effectively computable parametric expressions for extremal polynomials in terms of linear series of Poincare.