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

We consider an Abel equation (∗)y′ = p(x)y2+q(x)y3 withp(x), q(x)—polynomials in x. A centre condition for this equation (closely related to the classical centre condition for polynomial vector fields on the plane) is that y0 = y(0) ≡ y(1) for any solution y(x). This condition is given by the vanishing of all the Taylor coefficients vk(1) in the development y(x) = y0 + ∑∞ k=2 vk(x)y k 0 . Following Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear) we introduce periods of the equation (∗) as those ω ∈ C, for which y(0) ≡ y(ω) for any solution y(x) of (∗). The generalized centre conditions are conditions on p, q under which given a1, . . . , ak are (exactly all) the periods of (∗). A new basis for the ideals Ik = {v2, . . . , vk} has been produced in Briskin et al (1998 The Bautin ideal of the Abel equation Nonlinearity 10), defined by a linear recurrence relation. Using this basis and a special representation of polynomials, we extend results of Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), proving for small degrees of p and q a composition conjecture, as stated in Alwash and Lloyd (1987 Nonautonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129–52), Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), Briskin et al (Center Conditions II: Parametric and Model Centre Problems to appear). In particular, this provides transparent generalized centre conditions in the cases considered. We also compute maximal possible multiplicity of the zero solution of (∗), extending the results of Alwash and Lloyd (1987 Non-autonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129–52). PACS numbers: 34A34, 34A20