Local Center Conditions for Abel Equation and Cyclicity of its Zero Solution

An Abel di erential equation y p x y q x y is said to have a center at a pair of complex numbers a b if y a y b for any its solution y x with the initial value y a small enough Let p q be polynomials and let P R p Q R q P and Q satisfy Polynomial Composition condition if there exist polynomials P Q and W such that P x P W x Q x Q W x and W a W b The main result of this paper is that for a xed polynomial p satisfying some minor genericity restrictions and for a xed degree d of a polynomial q there exists p d such that for any polynomial q of degree d with the norm of q at most p d the Abel equation above has a center if and only if the Polynomial Composition condition is satis ed On this base we also provide an upper bound for the cyclicity of the zero solution of the Abel equation i e for the maximal number of periodic solutions which can appear in a small perturbation of the zero solution This research was supported by the ISF Grant No and by the BSF Grant No