Solution of the polynomial moment problem

TL;DR

This paper provides a comprehensive solution to the polynomial moment problem, which involves characterizing polynomials Q(z) orthogonal to all powers of a given polynomial P(z) over a segment, with applications to the center-focus problem.

Contribution

It offers a complete characterization of polynomials Q(z) orthogonal to powers of P(z), advancing understanding of the polynomial moment problem linked to Poincare's center-focus problem.

Findings

Characterization of all polynomials Q(z) orthogonal to powers of P(z).

Connection established between the polynomial moment problem and Poincare's center-focus problem.

Resolution of a problem posed over 10 years ago in polynomial orthogonality.

Abstract

In this paper we give a complete solution of the following "polynomial moment problem" which arose about 10 years ago in connection with Poincare'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].