Center conditions, compositions of polynomials and moments on algebraic curves

We consider an Abel equation (*) y' = p(x)y(2) + q(x)y(3) with p(x), q(x) polynomials in x. A center condition for (a) (closely related to the classical center condition for polynomial vector fields on the plane) is that y(0) = y(0) = y(1) for any solution y(x) of(*). This condition is given by the vanishing of all the Taylor coefficients v(k)(1) in the development y(x) = y(0) + Sigma(k=2)(infinity) v(k)(x)y(0)(k) Anew basis for the ideals I-k = {v(2),...,v(k)} has recently been produced, defined by a linear recurrence relation. Studying this recurrence relation, we connect center conditions with a representability of P = integral p and Q = integral q in a certain composition form (developing further some results of Alwash and Lloyd), and with a behavior of the moments integral P-k q. On this base, explicit center equations are obtained for small degrees of p and q.