Simplified Versions of Hilbert 16th Problem and Related Topics in Complex Dynamics and Analytic Foliations

该项目涉及与希尔伯特的第16个问题有关的平面微分方程式理论中的许多主题，并继续之前的两个项目。它既考虑实方程，也考虑复方程。在前一个项目的基础上，本项目的工作取得了两项主要成果。首先，相当出人意料的是，证明了实平面和复平面的多项式自同构的Kupka-Smer(KS)性质。证明基于“持久性定理”和“Petrovski-Landis(PL)策略”。一般而言，持久性定理要求多项式动力系统的某些几何性质在整个参数空间上全局扩张的可能性。PL策略利用持久性定理来证明或反证这些几何性质。给出了复空间中多项式自同构异宿点的新的持久性定理。作为推论，提出了此类自同构的Kupka-Smear性质的泛性。主要的工具将是PL战略。另一个结果是Glutsyuk和PI对实域和复域上的阿贝尔积分零点个数的一个上界估计。这一估计在雅科文科和他的学生的其他同类估计中是最好的。另一方面，它为无限小Hilbert第16问题的受限形式的完全解提供了一种途径。结合PL策略，给出了无穷小Hilbert 16问题本身的一种解决方法：给出了平面上一多项式的积分在另一多项式的椭圆上的实零点个数的一个上界。该项目提出了许多关于多项式动力系统的持久性、多项式叶的同时均匀化和拓扑性的问题。研究这些理论分支之间的关系是该项目的重要组成部分。动力系统理论分为两个部分：多维系统(混沌领域)和二维系统(有序领域)。希尔伯特16问题是二维系统理论中的一个核心问题。这个问题本身就是数学家们一百多年来所做的努力。与希尔伯特16号问题相关的百年研究历史在2002年发表在AMS公报上的一篇由PI发表的调查文章中进行了回顾。该调查特别包含了以前NSF项目的许多结果，以及受当前项目影响的问题。请注意，二维动力系统为物理、工程和生物学中的各种问题提供了模型(捕食者-被捕食者模型)。因此，理解真实的二维动力学是一个普遍的科学兴趣课题。另一方面，对真实动力系统的复杂扩张的研究提供了关于真实系统的重要新信息，而且本身就很有趣。

The project deals with numerous topics in the theory of planar differential equations related to Hilbert's 16th problem and continues two previous projects. It considers both real and complex equations. There are two major achievements in the work over the previous project that will be developed in the current one. First, rather unexpectedly, the Kupka-Smale (KS) property was proved for polynomial automorphisms both of real and complex planes. The proof is based on "persistence theorems" and "Petrovski-Landis (PL) strategy". Generally speaking, persistence theorems claim the possibility of global extension of some geometric properties of polynomial dynamical systems over the whole parameter space. PL strategy makes use of persistence theorems to prove or disprove such geometric properties. New persistence theorems for heteroclinic points of polynomial automorphisms of a complex space are expected. Genericity of Kupka-Smale property for such automorphisms is suggested as a consequence. The main tool would be the PL strategy. Another achievement is an upper estimate by Glutsyuk and the PI of the number of zeros of Abelian integrals both in real and complex domains. This estimate is the best amidst other estimates of this kind due to Yakovenko and his students. On the other hand, it provides an approach to the complete solution of the restricted version of the Infinitesimal Hilbert 16th Problem. Together with the PL strategy, this gives an approach to the Infinitesimal Hilbert 16th Problem itself: give an upper bound of the number of real zeros of an integral of a polynomial one-form over the ovals of another polynomial in the plane. The project suggests numerous problems on the persistence properties for polynomial dynamical systems, simultaneous uniformization and topological properties of polynomial foliations. Study of the relations between these branches of the theory is an important part of the project. Moreover, new simultaneous uniformization theorems, together with new generic properties of polynomial and analytic foliation of the complex space are expected.Theory of dynamical systems is split into two parts: multidimensional systems (realm of chaos); two-dimensional systems (realm of order). Hilbert 16th problem is a central one in the theory of two-dimensional systems. The problem itself persists the efforts of mathematicians during more than a hundred years. Centennial history of investigations related to Hilbert 16th problem is reviewed in a survey article by the PI published in the Bulletin of the AMS in 2002. The survey contains, in particular, many results of the previous NSF projects, as well as problems that are subject to the current project. Note that two-dimensional dynamical systems provide models for various problems in physics, engineering and biology (predator-prey models). Understanding of real two-dimensional dynamics is therefore a subject of general scientific interest. On the other hand, study of complex extensions of real dynamical systems provides important new information about real systems and is interesting in itself.