On the trigonometric moment problem

TL;DR

This paper investigates the classification of trigonometric polynomials orthogonal to powers of another polynomial, providing a natural solution under monodromy group conditions, especially for degrees up to 15 in the real case.

Contribution

It offers a simple, natural solution to the trigonometric moment problem based on monodromy group conditions, with explicit results for degrees up to 15 and exceptions up to degree 30.

Findings

Conditions hold for all real polynomials of degree ≤15.

Few exceptional monodromy groups exist up to degree 30.

Counter-examples are constructed for certain exceptional cases.

Abstract

The trigonometric moment problem arises from the study of one-parameter families of centers in polynomial vector fields. It asks for the classification of the trigonometric polynomials $Q$ which are orthogonal to all powers of a trigonometric polynomial $P$. We show that this problem has a simple and natural solution under certain conditions on the monodromy group of the Laurent polynomial associated to $P$. In the case of real trigonometric polynomials, which is the primary motivation of the problem, our conditions are shown to hold for all trigonometric polynomials of degree 15 or less. In the complex case, we show that there are a small number of exceptional monodromy groups up to degree 30 where the conditions fail to hold and show how counter-examples can be constructed in several of these cases.