Massimo Villarini

Some properties of planar polynomial systems of even degree

SummaryWe study autonomous systems in the plane of polynomial type. We obtain conditions for the existence of unbounded trajectories of such systems. As a consequence we prove that it does not exist a planar polynomial system of even degree with a global center.

Algebraic nonsolvability of the problem of existence of holomorphic first integrals

We prove that the existence of a holomorphic first integral of an analytic differential equation is an algebraically nonsolvable problem. Moreover, we prove a perturbation lemma showing that a nilpotent singularity of a differential equation in the plane cannont have a holomorphic first integral independently of the prolongation of anynth-order jet. We give an application to the case of nilpotent centers.

A Geometric Approach to the Existence of Sets of Periodic Orbits

We study the problem of the existence and of the geometric structure of the set of periodic orbits of a vector field in presence of a first integral. We give a unified treatment and a geometric proof of existence results of periodic orbits by Moser (local case) and Bottkol (global case) under a suitable nonresonance condition. The local resonance case is considered, too. For analytic vector fields admitting an analytic first integral, we give a geometric description of the set of periodic orbits, proving that it is an analytic set, hence extending a theorem by Siegel.