It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szeg\"o polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szeg\"o polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients {\it etc.} are derived.

Bivariate polynomials with a fixed leading term $x^m y^n$, which deviate least from zero in the uniform or $L^2$-norm on the unit disk $D$ (resp. a triangle) are given explicitly. A similar problem in $L^p$, $1 \le p \le \infty$, is studied on $D$ in the set of products of linear polynomials.