On Center Sets of ODEs Determined by Moments of their Coefficients

The classical H. Poincar\'{e} Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a {\em center}. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coefficients of the field. In this paper we show that the set of centers in the Center-Focus problem can be determined as the set of zeros of some continuous functions from the moments of coefficients of this equation.