Global Geometry of planar vector fields

这一提议是关于平面多项式向量场的。在一个多世纪之后，这些系统中仍有几个难题没有得到解决：庞加莱提出的两个问题和希尔伯特提出的一个问题，即他的第16个问题的第二部分。这项提议的研究有望为这些悬而未决的问题提供线索，并将对应用产生影响。以下几点表明了本项目的主要研究方向。***I)多项式向量场族的全局研究。一类二次微分系统是5维模作用的仿射变换和时间重标。到目前为止，从真正的全局观点出发，只研究了最多三个维度的子族，其中包括Lotka-Volterra系统的三维族，这对应用非常重要，由申请人与N. Vulpe一起研究。我们还将研究其他一些重要的子族，其中包括具有一阶弱焦点的四维二次向量场族，这对希尔伯特第16问题很重要。一些立方族也将被研究。***II)申请人将与合作者继续对整个二次类的奇点分类进行研究。她与art<s:1>， Llibre和Vulpe已经发表了4篇关于这个主题的论文，另一篇论文已经被接受发表，并于2014年7月提交了一篇论文。还有6种情况需要研究，并构成本研究计划的一部分：有四个不同的实有限奇点的情况，有四个不同的复奇点的情况，有两个实奇点和两个复奇点的情况，有3个实奇点，其中一个是双奇点，有两个复简单奇点和一个实双奇点，有一个多重奇点的情况。对于二次类，期望有超过1000个全局奇点构型。这个相当庞大的分类定理将会有应用。实际上，在涉及二次系统的应用数学模型的研究中，我们的算法可以用于有效地计算所涉及系统的全局奇点构型，只需按一下按钮即可获得信息。***III)另一个研究方向是多项式向量场在不变代数曲线上的可积性问题。poincarcarve关于代数可积性的问题和Darboux关于代数不变曲线的可积性的工作所引起的问题是开放的。Darboux（或Jouanolou）定理仅给出了Darboux（分别为代数）可积性的充分条件，且所涉及的曲线数不是最优的。申请人想要检测代数或达布可积性的必要条件，涉及曲线的度和多重性，曲线相对位置的条件，如曲线的相交多重性和曲线的奇点条件。**

This proposal is on planar polynomial vector fields.  There are several hard problems on these systems  still unsolved after more than a century: two problems posed by Poincaré and one posed by Hilbert, the second part of his 16th problem. The proposed research is expected to throw light on these open problems and will also have an impact on applications.  The following points indicate the main research lines of this project.  ***I) Global studies of families of polynomial vector fields.  The class of quadratic differential systems is 5-dimensional modulo the action of the group of affine transformations and time rescaling.  So far only sub-families of dimension at most three have been studied from a truly global viewpoint, among them the 3-dimensional family of Lotka-Volterra systems, very important for applications, studied by the applicant together with N. Vulpe. Some other important sub-families will be studied, among them the 4-dimensional family of quadratic vector fields with a first order weak focus, important  for Hilbert's 16th problem. Some cubic families will also be studied.***II) The applicant will continue her study with her collaborators on the classification of singularities of the whole quadratic class. Together with Artés, Llibre and Vulpe she already published 4 papers on this topic, another paper has been accepted for publication and another one was submitted in July 2014.  There remain 6 cases to be studied and form part of this research proposal: the case with four distinct real finite singularities, with 4 distinct complex singularities, with two real and two complex singularities, with 3  real singularities, one of them double, with two complex simple and one real double singularities and with a single real singularity of multiplicity four. Over 1000 global configurations of singularities for the quadratic class are expected. This rather huge classification theorem will have applications. Indeed, in the study of models in  applied mathematics involving quadratic systems, our algorithm could be used for effective computation of the global configurations of the singularities of the systems involved, information  available at the touch of a button.***III) Another line of research is on problems of integrability of polynomial vector fields in terms of invariant algebraic curves. Poincaré's problem on algebraic integrability and problems resulting from the work of Darboux on integrability in terms of algebraic invariant curves are open.  The Theorem of Darboux (or of Jouanolou) gives only sufficient conditions for Darboux (respectively algebraic) integrability and the number of curves involved is not optimal. The applicant wants to detect necessary conditions for algebraic or Darboux integrability, involving the degrees and multiplicities of the curves, conditions on the their relative position such as  intersection multiplicities of the curves and conditions on their singularities.   **