Hilbert's 16^th problem (the second part) states that the number of limit cycles of a polynomial vector field in the plane is bounded uniformly in the degree of the field. At first glance, this looks like a uniform finiteness statement, since the family of all polynomial vector fields of a fixed degree is definable in the real field. However, the corresponding family of limit cycles is not first-order definable in the real field; they are highly transcendental objects that can only be counted using something called Poincaré first-return maps.

I will explain a bit of the history of the problem and its supposedly easier cousin, Dulac's problem (which only claims finiteness without the uniformity), and I will explain what these first-return maps are. Our conjecture is that these (families of) first-return maps are definable in some large o-minimal expansion of the real field. I will explain how this conjecture implies Dulac's, and Hilbert's 16^th, problem, and I will give an idea of what we have managed to prove so far (with many collaborators who will be mentioned during the talk).