ON THE TRIGONOMETRIC MOMENT PROBLEM

AMELIA ALVAREZ, JOS E LUIS BRAVO, COLIN CHRISTOPHERAbstract. The trigonometric moment problem arises from the studyof one-parameter families of centers in polynomial vector elds. It asksfor the classi cation of the trigonometric polynomials Q which are or-thogonal to all powers of a trigonometric polynomial P.We show that this problem has a simple and natural solution undercertain conditions on the monodromy group of the Laurent polynomialassociated to P. In the case of real trigonometric polynomials, which isthe primary motivation of the problem, our conditions are shown to holdfor all trigonometric polynomials of degree 15 or less. In the complexcase, we show that there are a small number of exceptional monodromygroups up to degree 30 where the conditions fail to hold and show howcounter-examples can be constructed in several of these cases.