We investigate semiconjugate rational functions, that is rational functions A, B related by the functional equation A • X = X • B, where X is a rational function. We show that if A and B is a pair of such functions, then either A can be obtained from B by a certain iterative process, or A and B can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.

In the paper [24] Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly polynomial solutions of the functional equation f (p(z)) = g(q(z)).In this paper we study the equation above in the case when f, g, p, q are holomorphic functions on compact Riemann surfaces. We also construct a selfcontained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.

Abstract. In this paper we give a solution of the following "polynomial moment problem" which arose about ten years ago in connection with Poincaré's center-focus problem: for a given polynomial P (z) to describe polynomials q(z) orthogonal to all powers of P (z) on a segment [a, b].

We consider Cauchy type integrals I(t) = 1 2πi γ g(z)dz z−t with g(z) an algebraic function. The main goal is to give constructive (at least, in principle) conditions for I(t) to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the Monodromy group of the algebraic function g, the geometry of the integration curve γ, and the analytic properties of the Cauchy type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy type integrals of algebraic functions appear in the infinitesimal versions of two classical open questions in Analytic Theory of Differential Equations: the Poincaré Center-Focus problem and the second part of the Hilbert 16-th problem.

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