The Hilbert 16-th Problem and an Estimate for Cyclicity of an Elementary Polycycle

One way to formulate the Hilbert 16-th Problem is the following: Hilbert 16-th Problem (HP). Find an estimate for H(n) for any n ∈ Z+. We shall discuss problems related to the following: Existential Hilbert 16-th Problem (EHP). Prove that H(n) < ∞ for any n ∈ Z+. The problem about finiteness of number of limit cycles for an individual polynomial line field (1) is called Dulac problem since the pioneering work of Dulac who claimed in 1923 to solve this problem, but an error was found by Ilyashenko. The Dulac problem was solved by two independent and rather different proofs given almost simultaneously by Ilyashenko [I] and Ecalle [E]. However, both proofs do not allow any generalization to solve Existential Hilbert Problem. Consider the equation (1) for different polynomials (Pn(x, y), Qn(x, y)) as the family of line fields on R depending on parameters of the polynomials. Using a central projection π : S → R and homogenuity with respect to parameters of the equation (1) (line fields λPn(x, y)/λQn(x, y) and Pn(x, y)/Qn(x, y) for any λ 6= 0 are the same) one can construct a finite parameter family of analytic line fields on the shpere S with a compact parameter base B (see e.g. [IY2] for details). After this reduction Existential Hilbert Problem becomes a particular case of the following Global Finiteness Conjecture (GFC). (see e.g. [R]) For any family of line fields on S with a compact parameter base B the number of limit cycles is uniformly bounded over all parameter values. We refer the reader to the volumes [S] and [IY2] where various development of these and related problems are discussed. Families of analytic fields are extremely difficult to analyze. In the middle of 80’s Arnold [AAI] proposed to consider generic families of smooth vector fields on S. A smooth analog of Global Finiteness Conjecture is the following