Hilbert Type Numbers and Related Topics in Analytic Differential Equations

建议：DMS-9970372首席研究员：Yulij Ilyashenko摘要：希尔伯特在他的第16个问题中问道：“关于给定次数的平面多项式向量场的极限环的数量和位置可以说什么？这道题的主要说明是：“给出上述向量场极限环个数的一个上界。”极限环的最大可能个数在这种情况下被称为“希尔伯特数”。“这个数的存在性甚至对于二阶向量场也没有得到证明。Arnold、Ilyashenko、Pugh等人提出的问题存在各种类似物和简化。这些简化版本中的每一个都需要对某些特定类别的微分方程的极限环的最大可能数目的上限估计。这个类依赖于一个参数，该参数通常是类定义中涉及的某个多项式的次数。所需的量被称为“希尔伯特型数”。“本项目涉及阿贝尔和Lienard方程，二次向量场和阿贝尔积分的零点。希尔伯特第16问题的相关简化版本在过去的三十年中被提出，但除了极少数例外，它们也仍然没有解决。这个项目的主要目标是给出一些“希尔伯特型数”的上估计，不是针对定义这些数的整个类，而是针对一些简化的方程组。一方面，这些估计应该大大扩展我们目前的知识。另一方面，他们将有望提供工具，找到明确的上限估计类似的希尔伯特数在其原始setting. In开始的这个世纪，希尔伯特提出了一个清单的问题，旨在预测发展的数学过程中，随后的百年。现在，在世纪末，很明显他的预言是正确的。在大多数情况下，希尔伯特问题已经解决，他们的解决方案是在最近的数学史上的亮点。然而，有些问题仍未解决。其中之一是希尔伯特第16问题的第二部分。 它要求对平面上多项式微分方程的一些定量几何特征给出一个上估计。 这些特征被称为“希尔伯特数”。“尽管世纪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，解决问题的方法必须通过简化问题来进行。简化的问题需要估计相同的几何特征，但类的方程是不太复杂的比原来的希尔伯特问题。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广大读者可以理解的图片来说明。另一方面，解可能涉及代数几何、复分析、拓扑学等不同数学领域的深入结果。平面微分方程理论中发展起来的方法在数学与其他学科的边界上找到了许多应用，如突变理论、应用数学、生物建模等。因此，这一主题对年轻数学家很有吸引力，其中一些人（包括主要研究者的学生）对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.