Global geometry of families of polynomial vector fields

我与合作者art<s:1>、Llibre和Vulpe一起，经过8年的努力，得到了平面二次向量场QS类奇点全局构型的几何分类，得到了1765个这样的构型。我们的书（680页）将于今年出现在Birkhäuser b施普林格系列丛书中。介绍了多项式向量场的几何等价关系。这种关系比拓扑关系更深刻，它还考虑了奇点的代数和几何特征，如弱奇点的阶数、奇点的多重性、等时性水平等。我们后来获得了QS奇异点的全局构型的拓扑分类，产生了208个构型，预计将发表在quality . theory of Dyn. system的2020年第1期。这为获得所有QS的拓扑分类——模极限环——开辟了道路。事实上，奇点的每一个全局拓扑构型都构成了一个骨架，在这个骨架上，基于可能连接的组合研究，可以构造拓扑相图模极限环。这可能需要几年的时间。我们在书中介绍的多项式向量场的一般理论框架，为研究多项式系统族开辟了新的道路。特别地，我们打算研究具有1阶弱焦点的QS系统的族QW1。弱焦点在极限环的产生中是非常重要的，因此对于希尔伯特的第16个问题（H16）也是如此。该类是四维的，模仿射群的作用和时间重标，目前还没有研究QS的四维子类。（Llibre、art<s:1>和我研究了QS中第3和第2弱灶的2维和3维族QW3和QW2。）申请人亦建议研究建议书中提到的几个三次系统族。我的另一个研究方向是Darboux，代数和Liouvillian可积性。Darboux只给出了Darboux可积性的充分条件。庞加莱卡尔关于识别一个系统何时是代数可积的问题（即具有一个有理的第一积分），即使对于QS中的系统仍然是开放的。我计划与我的博士生Ana Maria Travaglini和我的合作者Regilene Oliveira一起，从可积性的角度研究具有不变双曲线的QS系统的QSH族。这个家族展示了丰富多样的几何结构。这是一个很好的试验场，可以观察系统的几何性质和动态性质是如何融合在一起产生不同类型的可积性的。这项工作是在代数几何和动力系统的界面。最后，申请人和Llibre打算为研究QS中的图形族提供良好的几何背景（包括所需的几何等价关系），然后将其应用于H16的存在部分。

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.