An explicit bound of the number of vanishing double moments forcing composition

We give two new characterizations of pairs of polynomials or trigonometric polynomials that form a composition pair. One of them proves that the cancellation of a given number of double moments implies that they form a composition pair. This number only depends on the maximum degree of both polynomials. This is the first time that composition is characterized in terms of the cancellation of an explicit number of double moments. Our results allow to recognize the composition centers for polynomial and trigonometric Abel differential equations.