Generalized centre conditions and multiplicities for polynomial Abel equations of small degrees

We consider an Abel equation(*)y' = p(x)y(2) +q(x)y(3) with p(x), q (x)-pol ynomials in x. A centre condition for this equation (closely related to the classical centre condition for polynomial vector fields on the plane) is t hat y(0) = y(0) equivalent to y(1) for any solution y(x). This condition is given by the vanishing of all the Taylor coefficients v(k)(1) in the devel opment y(x) = y(0) + Sigma(k=2)(infinity) v(k)(x)y(0)(k). Following Briskin el al (Centre Conditions, Composition of Polynomials and Moments on Algebr aic Curves to appear) we introduce periods of the equation (*) as those ome ga epsilon C, for which y(0) equivalent to y(omega) for any solution y(x) o f (*). The generalized centre conditions are conditions on p, q under which given a(1),..., a(k) are (exactly all) the periods of (*). A new basis for the ideals I-k = (v(2),...,v(k)) has been produced in Briskin et al (1998 The Bautin ideal of the Abel equation Nonlinenrity 10), defined by a linear recurrence relation. Using this basis and a special representation of poly nomials, we extend results of Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), proving for sma ll degrees of p and q a composition conjecture, as stated in Alwash and Llo yd (1987 Nonautonomous equations related to polynomial two-dimensional syst ems Proc. R. Soc. Edinburgh A 105 129-52), Briskin et al (Centre Conditions , Composition of Polynomials and Moments on Algebraic Curves to appear), Br iskin er al (Center Conditions II: Parametric and Model Centre Problems to appear). In particular, this provides transparent generalized centre condit ions in the cases considered. We also compute maximal possible multiplicity of the zero solution of (*), extending the results of Alwash and Lloyd (19 87 Non-autonomous equations related to polynomial two-dimensional systems P roc. R. Sec. Edinburgh A 105 129-52).