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列表中最持久的问题之一，在这方面仅次于黎曼假设。本课题主要研究二次向量场的希尔伯特第十六次问题，希尔伯特第十六次问题的无穷小版本，以及几个相关的主题。后者包括：高维多项式微分方程的代数解；复多项式叶理的刚性；椭圆曲线模量与旋转数的关系保角映射胚芽解析族的新问题，与所谓混合族有关。该项目的第一部分是基于在首席研究员之前的奖励下获得的两个主要结果。第一个结果几乎完全解决了二次向量场的受限希尔伯特问题。此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一个“覆盖函数”，该覆盖函数具有以下性质：(1)它在没有多环（分离矩阵多边形）的向量场集合上是有限的下半连续的；(2)它对该集合的约束在具有多环的向量场的不相集上有无穷极限；(3)使排他集外的极限环数最大化。主要研究者提出通过建立极限环的局部持久性来寻找极限环数的上界，然后用一个仍然使极限环数最大化的截止函数代替覆盖函数。截止函数的最大值将是期望的上界。当前项目的第二个主要目标是受限无穷小希尔伯特十六次方问题的完全解。这个解涉及到另一个覆盖函数，这次是在给定次数的所有超莫尔斯多项式的集合上，它具有以下性质：它使任何多项式的阿贝尔积分的实数0的数量最大化小于给定次数的一个形式的实数0除以实数超莫尔斯多项式的实数椭圆。覆盖函数在非超莫尔斯多项式集合上具有极点。同样，我们的目标是证明阿贝尔积分的零的局部持久性定理，并通过用截断函数代替覆盖函数来推导这些零的个数的上界。这将解决无穷小的希尔伯特问题。后半部分所描述的问题（代数解、刚性、椭圆曲线的模、混合族）是微分方程、复分析和代数几何的结合。虽然它们是独立的，但至少有一半与希尔伯特的问题有关。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核心问题。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了模型（例如生物学中的捕食者-猎物模型）。因此，对真实二维动力学的理解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展的研究提供了关于真实系统的重要新信息，并且本身就很有趣。事实上，一些专家甚至说，“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.