On polynomials orthogonal to all powers of a given polynomial on a segment

In this paper we investigate the following “polynomial moment problem”: for a given complex polynomial P(z) and distinct a,b∈C to describe polynomials q(z) orthogonal to all powers of P(z) on [a,b]. We show that for given P(z), q(z) the condition that q(z) is orthogonal to all powers of P(z) is equivalent to the condition that branches of the algebraic function Q(P−1(z)), where Q(z)=∫q(z)dz, satisfy a certain system of linear equations over Z. On this base we provide the solution of the polynomial moment problem for wide classes of polynomials. In particular, we give the complete solution for polynomials of degree less than 10.