The aim of this paper is to consider analytic functions, arising in connection with algebraic differential equations, from the following point of view: Taylor coefficients of such functions usually turn out to be polynomials (or rational functions) in the parameters of the problem. Following pioneering work of Bautin [1], we relate analytic properties of the functions considered with algebraic properties of various ideals, generated by their Taylor coefficients. Bautin's approach has been successfully applied to the study of the Poincare mapping of the planar polynomial systems in [4,9,27], and to the study of the period function of polynomial Hamiltonian systems in [5,6,12,13].
However, the study of the Poincare mapping of the polynomial differential equations (closely related to the second part of Hilbert 16th problem) is, probably, the most difficult among a wide variety of problems, to which Bautin's approach can be applied. In particular, counting zeroes of solutions of algebraic differential equations is an important and mostly open problem. To our best knowledge, the first application of Bautin's method to this problem was given in [26]. In the present paper we systematically apply this method in the following two situations: to algebraic functions and to solutions of the first order polynomial differential equations (higher order linear equations are also considered in Section 5).