In 1901 D. Hilbert formulated a problem that seemed to be at that time a minor digression outside of the realm of algebraic geometry. Discussing the algebraic question about real ovals of planar algebraic curves, he asked in passing, whether a similar question about oval-like solutions of a planar polynomial ODE can be addressed by his proposed "deformation method".

More than hundred years later we still do not have a clear understanding of what the Hilbert's question is about (not to mention that we do not know the answer to it except for rather limited finiteness results). In my lectures I will try to argue that there exists a (necessarily limited) possibility to extend results (but not methods!) of algebraic geometry to functions defined by polynomial differential equations, ordinary and Pfaffian. One of the results obtained on this way implies a constructive solution to a so called infinitesimal Hilbert problem, related to deformations of integrable ODEs.

Le lieu d'annulation dans$\R^n$ d'un polynôme de degré d en n variables est en général une hypersurface lisse qui possède de nombreuses composantes connexes detopologies variées. À quelle topologie s'attendre lorsque l'on choisit un polynôme au hasard ? Cette question sera le thème principal de mon cours et j'expliquerai quelques-uns des résultats que l'on a pu obtenir avec Damien Gayet. On établit en particulier des majoration et minoration des nombres de Betti attendus de cette hypersurface, pour les grandes valeurs de d.