Global geometry of families of polynomial vector fields

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Together with my collaborators Artés, Llibre and Vulpe we obtained, after 8 years of work, the geometric classification of the global configurations of singularities of the class QS of planar quadratic vector fields, yielding 1765 such configurations. Our book (680 pp.) is due to appear in the Birkhäuser series of Springer this year. We introduced the geometric equivalence relation for polynomial vector fields. This relation is deeper than the topological one, taking into account also algebraic and geometric features of singularities such as the order of weak singularities, multiplicity of singularities, level of isochronicity, etc. We later obtained the topological classification of the global configurations of singularities of QS yielding 208  configurations, due to appear in in the first issue of 2020 of Qual.Theory of Dyn. Syst. This opened the road for obtaining the topological classification, modulo limit cycles, of all QS. Indeed, each global topological configuration of singularities constitutes a skeleton over which the topological phase portraits modulo limit cycles could be constructed, based on a combinatorial study of possible connections. This may take a several years. The general theoretical framework we introduced for polynomial vector fields in our book, opened new roads for studying families of polynomial systems. In particular we intend to study the family QW1 of systems in QS possessing a weak focus of order 1. Weak foci are very important in the production of limit cycles and hence for Hilbert's 16th problem (H16). This class is 4-dimensional, modulo the action of the affine group and time rescaling and so far no 4-dimensional subclass of QS was studied. (The 2 and 3-dimensional families QW3 and QW2 of families in QS with 3rd and 2nd weak foci were studied by Llibre, Artés and myself.) The applicant also proposes to study several families of cubic systems mentioned in the proposal.  Another direction of my research is on Darboux, algebraic and Liouvillian integrability. Darboux gave only sufficient conditions for Darboux integrability.  Poincaré's problem on recognizing when a system is algebraically integrable (i.e. having a rational first integral), is still open even for systems in QS. Together with my Ph.D. student Ana Maria Travaglini and my collaborator Regilene Oliveira, we plan to study from the viewpoint of integrability, the family QSH of systems in QS having an invariant hyperbola. This family displays a rich variety of geometric structures. It is a good testing ground for seeing how the geometric and dynamic properties of the systems blend in producing the different kinds of integrability. This work lies at the interface of algebraic-geometry and dynamical systems. Finally the applicant together with Llibre intend to give a sound geometrical background (including a needed geometrical equivalence relation) for studying the family of graphics in QS and then apply it to the existential part of H16.