Sala Riunioni (Dip. Matematica).
The infinitesimal Hilbert problem addresses limit cycles of planar polynomial vector field, which appear by a small non-conservative perturbation of an integrable (Hamiltonian) system. Their number is closely related to the number of algebraic level ovals of a bivariate polynomial corresponding to zero periods of a polynomial 1-form. I will describe a class of special transcendental functions (of which the periods are a special particular case) which resemble algebraic functions in the sense that they admit explicit upper bounds for the number of isolated roots in the real and complex domain. These bounds give the answer to the Infinitesimal Hilbert problem.