Infinitesimal Hilbert 16th problem

I will describe the current state of affairs in both the original Hilbert 16th problem 

(on limit cycles of polynomial planar vector fields) and its relaxed version on zeros of 

Abelian integrals. It turns out that the latter belong to a natural class of Q-functions 

described by integrable systems of linear differential equations with quasiunipotent monodromy, 

defined over the field of rational numbers. Functions of this class admit explicit (albeit very 

excessive) bounds for the number of their isolated zeros in a way similar to algebraic functions. 

This result lies at the core of the solution of the infinitesimal Hilbert problem, achieved with 

Gal Binyamini and Dmitry Novikov.

 

The talk is aimed at a broad audience.