Generalized "second Ritt theorem" and explicit solution of the polynomial moment problem

Abstract

In the recent paper arXiv:0710.4085 was shown that any solution of "the polynomial moment problem", which asks to describe polynomials Q orthogonal to all powers of a given polynomial P on a segment, may be obtained as a sum of some "reducible" solutions related to different decompositions of P into a composition of two polynomials of lesser degrees. However, the methods of arXiv:0710.4085 do not permit to estimate the number of necessary reducible solutions or to describe them explicitly. In this paper we provide a description of the polynomial solutions of the functional equation P=P_1(W_1)=P_2(W_2)=...=P_r(W_r), and on this base describe solutions of the polynomial moment problem in an explicit form suitable for applications. With respect to the previous version a more general form of the generalized "secon Ritt theorem" is proved and the proof is considerably simplified. Besides, a missed case in Theorem 1.2 was added and the proof is corrected.