THE BAUTIN IDEAL OF THE ABEL EQUATION

An approach to the centre-focus problem for homogeneous perturbations proposed by Cherkas yields a transformation to periodic Abel equations of degree 3. In this paper we consider both the polynomial and period ic Abel equations of any degree. We define the Bautin ideal for these two classes of Abel equations. Recently an approach based on the use o f a 1-parameter integrating factor allowed to find the successive deri vatives of the return map for a polynomial system which is a homogeneo us perturbation of the rotation at the origin. We present the same typ e of results for the Abel equations. For the polynomial Abel equations , we show that there is an integrating factor defined by a convergent series expansion with polynomial coefficients which satisfy a simple l inear recurrency relation. We solve this recurrency relation for low d egrees of the perturbation and compute the Bautin index. We then use o ur previous findings based on the Bernstein inequality and Bautin inde x to bound the number of complex periodic solutions on a neighbourhood of prescribed size. For the periodic Abel equations, we show that the existence of the integrating factor is equivalent to the periodicity of all other orbits.