A simple solution of some composition conjectures for Abel equations

Trigonometric Abel differential equations appear in the study of the number of limit cycles and the center-focus problem for certain families of planar polynomial systems. The composition centers are a class of centers for trigonometric Abel equations which have been widely studied during last years. We characterize this type of centers as the ones given by couples of trigonometric polynomials for which all the generalized moments vanish. They also coincide with the strongly and the highly persistent centers. Our result gives a simple and self-contained proof of the so called Composition Conjecture for trigonometric Abel differential equations. We also prove a similar version of this result for Abel equations with polynomial coefficients.