AND INFINITESIMAL HILBERT 16TH PROBLEM

1. Hilbert problem as a paradigm The question on the maximal number (and position) of limit cycles of a planar polynomial vector field, the famous second part of the Hilbert 16th problem, remains open despite more than one century of continuing efforts. One possible explanation of why this problem is so unyielding, may look as follows. The enumeration problem that is perfectly legal with respect to algebraic objects (e.g., ovals of real algebraic curve, actually the first part of the same 16th problem), becomes artificial when applied to utterly transcendental objects like solutions of polynomial differential equations. The same phenomenon can be illustrated as follows. The question about the number of isolated zeros (real or complex) makes perfect sense for uni-variate polynomials (the answer is widely known). The only modification required for algebraic functions of one variable is to take care of the mul-tivaluedness (e.g., making slits and choosing branches). Yet when instead of algebraic functions the same question is raised concerning transcendental functions like the Poincare ́ displacement function (the difference between the Poincare ́ map and the identity), the problem becomes transcendentally hard. Needless to say, this example is the same Hilbert problem. There exist several relaxations of the Hilbert problem, all of them differ-ent localizations. This means that either only part of limit cycles is to be counted (say, born by small perturbations from a singular point or a sep-aratrix polygon), or only fraction of polynomial vector fields is considered (say, systems obtained by small perturbation from integrable planar vector fields that by definition have no limit cycles). The latter localization admits one further simplification (in a sense, linearization). Let H(x, y) be a poly-nomial in two real variables (the Hamiltonian) and P,Q two other bivariat