Hilbert Type Numbers and Related Topics in Analytic Differential Equations

摘要：Hilbert在他的第16个问题中问道：“关于给定度的平面多项式向量场的极限环的数目和位置，我们可以说些什么？”这个问题的主要说明是：“给出上述向量场的极限环数的上界。”在这种情况下，极限环的最大可能数被称为“希尔伯特数”。即使对于二阶向量场，这个数的存在性也尚未得到证明。Arnold、Ilyashenko、Pugh等人对这个问题提出了各种类比和简化。这些简化版本中的每一个都需要对某些特定类型的微分方程的最大可能极限环数有一个上限估计。该类依赖于一个参数，该参数通常是类定义中涉及的某个多项式的度数。所需的数量被称为“希尔伯特型数”。本项目涉及阿贝尔方程和利纳德方程，二次向量场，以及阿贝尔积分的零点。希尔伯特第16个问题的相关简化版本在过去三十年中被提出，但除了极少数例外，它们也没有得到解决。这个项目的主要目标是给出“希尔伯特类型数”的一些上限估计，不是对定义这些数的整个类，而是对一些简化的方程集。一方面，这些估计将大大扩展我们目前的知识。另一方面，它们将有望提供工具，为希尔伯特数在原始设置下的类似物找到明确的上限估计。在本世纪初，希尔伯特提出了一系列问题，旨在预测数学在随后的一百年中的发展。现在，在本世纪末，他的预言显然是正确的。在很大程度上，希尔伯特问题已经得到了解决，它们的解决是近代数学史上的亮点。然而，有些问题仍未得到解决。其中包括希尔伯特第十六题的第二部分。它要求平面上多项式微分方程的一些定量几何特征的上估计。这些特征被称为“希尔伯特数”。尽管一个世纪以来杰出的数学家们付出了巨大的努力，但到目前为止还没有发现一个希尔伯特数。现在很清楚，解决问题的途径必须通过简化问题来进行。简化的问题需要对相同的几何特征进行估计，但对比原始希尔伯特问题更简单的方程进行估计。人们希望这些简化版本的解决方案将为解决原始问题提供工具。平面微分方程的美妙之处在于，这个理论中的问题和答案可以用通俗易懂的图片来说明。另一方面，解可能涉及不同数学领域的深层结果，如代数几何、复分析、拓扑等。在平面微分方程理论中发展的方法在数学与其他学科的边界上有许多应用，如突变理论、应用数学、生物建模等。因此，该学科对年轻的数学家很有吸引力，其中一些人（包括首席研究员的学生）在该领域做出了重大贡献。

Proposal: DMS-9970372Principal Investigator: Yulij IlyashenkoAbstract: Hilbert asked in his 16th problem: " What may be said about the number and location of limit cycles of planar polynomial vector fields of given degree?" The main specification of this question is: "Give an upper bound on the number of limit cycles for the above vector fields." The maximal possible number of limit cycles in this context is called a "Hilbert number." The existence of this number has not yet been proved even for degree two vector fields. There exist various analogs and simplifications of the problem proposed by Arnold, Ilyashenko, Pugh and others. Each of these simplified versions requires an upper estimate of the maximal possible number of limit cycles in some specific class of differential equations. This class depends on a parameter which is typically a degree of some polynomial involved in the definition of the class. The required quantities are called "Hilbert type numbers." The present project deals with Abel and Lienard equations, quadratic vector fields, and zeros of Abelian integrals. Related simplified versions of the Hilbert 16th problem were stated during last three decades, but with very few exceptions, they too remain unsolved. The main objective of this project is to give some upper estimates of "Hilbert type numbers," not for the whole class for which these numbers are defined, but for some reduced set of equations. On the one hand, these estimates should greatly extend our present knowledge. On the other hand, they will hopefully provide tools for finding explicit upper estimates for analogs of the Hilbert numbers in their original setting.At the beginning of this century, Hilbert presented a list of problems intended to predict the development of mathematics over the course of the ensuing hundred years. Now, at the end of the century, it is clear that his prediction was correct. For the most part, the Hilbert problems have been solved, and their solutions are highlights in the recent history of mathematics. Some of the problems, however, remain unsolved. Among these is the second part of Hilbert's 16th problem. It requires an upper estimate for some quantitative geometric characteristics of polynomial differential equations in the plane. These characteristics are called "Hilbert numbers." Not a single Hilbert number has been found up to the present time, despite tremendous efforts by brilliant mathematicians throughout the century. It is clear now that the way to the solution must proceed through simplifications of the problem. Simplified problems require estimations for the same geometric characteristics, but for classes of equations which are less complicated than those in the original Hilbert problem. It is the hope that solutions to these simplified versions will furnish tools for attacking the original problem. The beauty of planar differential equations is that questions and answers in this theory can be illustrated by pictures understandable to a broad audience. On the other hand, the solutions may involve deep results from different domains of mathematics like algebraic geometry, complex analysis, topology and so on. The methods developed in the theory of planar differential equations find numerous applications on the boundary of mathematics with other disciplines such as catastrophe theory, applied mathematics, biological modelling, etc. For this reason, the subject is attractive to young mathematicians, some of whom (including students of the Principal Investigator) have made significant contributions to the field.