Hilbert 16th Problem and Related Topics in Complex Analysis and Foliations

希尔伯特的第16个问题仍然是他著名的问题列表中最持久的问题之一，在这方面仅排在黎曼假设之后。这个项目的重点是二次向量场的希尔伯特第十六问题，希尔伯特第十六问题的无穷小版本，以及几个相关的主题。后者包括以下内容：代数解决方案的多项式微分方程在更高的层面;刚性复杂的多项式叶理;之间的关系模的椭圆曲线和旋转数;新的问题分析家庭芽的共形映射，有关所谓的混合家庭。该项目的第一部分是基于主要研究者以前获得的两项主要成果。第一个结果是二次向量场的限制希尔伯特问题的几乎完全解。该解决方案将在当前项目中完成。在其最终形式中，该解将提供二次向量场空间上的“覆盖函数”，该覆盖函数具有以下性质：（1）它在没有多圈的向量场集合上是有限的和下连续的（2）它对这一集合的限制在具有多圈的向量场的排他集合上具有无限的极限;（3）优化排他集外的极限环个数。主要研究者建议通过建立极限环的局部持久性属性，然后用仍然优化极限环数量的截止函数替换覆盖函数，来找到极限环数量的上限。截止函数的最大值将是所需的上限。当前项目的第二个主要目标是完全解决限制无穷小希尔伯特第十六问题。这个解决方案涉及到另一个覆盖函数，这一次的所有ultra-Morse多项式的给定程度，具有以下性质：它majorizes的数量真实的零的阿贝尔积分的任何多项式一个形式的程度小于给定的一个超过真实的椭圆的真实的ultra-Morse多项式。覆盖函数在非超莫尔斯多项式集上有极点。再次，我们的目标是证明一个局部持久性定理的零的阿贝尔积分，并推导出一个上限，这些零的数量取代覆盖函数与截止函数。这将解决无穷小希尔伯特问题。在建议的后半部分中所描述的问题（代数解、刚性、椭圆曲线的模量、混合族）位于微分方程、复分析和代数几何的界面上。动力系统理论分为两部分：多维系统（混沌领域）和二维系统（有序领域）。希尔伯特第十六问题是二维系统理论中的一个中心问题。众所周知，二维动力系统为物理学、工程学和生物学中的各种问题提供模型（例如，生物学中的捕食者-被捕食者模型）。因此，对真实的二维动力学的理解是具有普遍科学意义的课题。另一方面，研究真实的动力系统的复扩张提供了关于真实的系统的重要的新信息，并且其本身是有趣的。事实上，有些专家甚至说，《自然之书》是用复杂分析的语言写成的。

Hilbert's sixteenth problem remains among the most persistent in his famous list, yielding first place in this regard only to the Riemann Hypothesis. This project focuses on Hilbert's sixteenth problem for quadratic vector fields, the infinitesimal version of Hilbert's sixteenth problem, and several related topics. The latter include the following: algebraic solutions of polynomial differential equations in higher dimensions; rigidity of complex polynomial foliations; relations between moduli of elliptic curves and rotation numbers; new problems about analytic families of germs of conformal maps, related to so-called mixed families. The first part of the project is based on two major results obtained under the principal investigator's previous award. The first of those results is an almost complete solution to the restricted Hilbert problem for quadratic vector fields. This solution is to be completed in the current project. In its final form, this solution would provide a "covering function" on the space of quadratic vector fields, a covering function with the following properties: (1) it is finite and lower semicontinuous on the set of vector fields that have no polycycles (separatrix polygons); (2) its restriction to this set has an infinite limit on the exclusive set of vector fields with polycycles; (3) it majorizes the number of limit cycles outside the exclusive set. The principal investigator proposes to find an upper bound on the number of limit cycles by establishing a local persistence property for limit cycles and then replacing the covering function with a cut-off function that still majorizes the number of limit cycles. The maximum of the cut-off function will be the desired upper bound. The second major objective of the current project is the complete solution of the restricted infinitesimal Hilbert sixteenth problem. This solution involves another covering function, this time on the set of all ultra-Morse polynomials of given degree, with the following property: it majorizes the number of real zeros of an Abelian integral of any polynomial one-form of degree smaller than the given one over the real ovals of a real ultra-Morse polynomial. The covering function has poles on the set of non-ultra-Morse polynomials. Once more, the goal is to prove a local persistence theorem for zeros of Abelian integrals and to derive an upper bound for the number of these zeros by replacing the covering function with a cut-off function. This will solve the infinitesimal Hilbert problem. The problems described in the latter part of the proposal (algebraic solutions, rigidity, moduli of elliptic curves, mixed families) lie at the interface of differential equations, complex analysis, and algebraic geometry. While they are of independent interest, at least half of them are related to Hilbert's problem.The theory of dynamical systems is divided into two parts: multidimensional systems (the realm of chaos) and two-dimensional systems (the realm of order). Hilbert's sixteenth problem is a central one in the theory of two-dimensional systems. It is well known that two-dimensional dynamical systems provide models for various problems in physics, engineering, and biology (e.g., predator-prey models in biology). The understanding of real two-dimensional dynamics is therefore a subject of general scientific interest. On the other hand, the study of complex extensions of real dynamical systems provides important new information about real systems and is interesting in its own right. Indeed, some experts even say that the "Book of Nature" is written in the language of complex analysis.