Restricted Versions of the Hilbert 16th Problem and Related Topics in the Theory of Analytic Foliations

希尔伯特第16题，第2部分，是：“关于平面多项式向量场的极限环的数目和位置，可以说什么？”传统上，这个问题被解释为求多项式向量场的极限环个数的上界作为多项式次数的函数的问题。即使对于二次系统，其上界也还没有找到，它的存在性也没有得到证明。考虑希尔伯特16号问题的“受限版本”是有意义的。也就是说，集合或所有多项式向量场由相似类的特定子集代替。例如Lienard方程和Abel方程(后者是系数为1周期的相变量的多项式)。即使对于这些方程，极限环的数量问题也是悬而未决的。它是用PI加一个额外的限制来求解的：Abel方程和具有奇次多项式的Lienard方程与系数的幅值的上界一起考虑；极限环的数目的上限估计取决于这个幅值的上界。这是之前NSF支持的结果。在本项目中，我们试图摆脱后一种限制，并给出仅依赖于右侧多项式的次数的极限环个数的估计；对于Abel方程，系数应该是给定次数的三角多项式。我们希望利用复解析叶理论、全纯函数的增长性和零点定理等强大的工具，以及在NSF支持下发展起来的方法。另一个需要研究的重要问题是无穷小的希尔伯特第16问题。它需要估计由哈密顿多项式向量场的椭圆产生的小扰动所产生的极限环的个数；后者的闭合轨道形成连续族。这个问题归结为对阿贝尔积分零点个数的估计，即平面上实多项式椭圆上的多项式1形式的积分；估计应以多项式哈密顿函数的次数和被积函数的次数给出。作者从69年开始研究这个问题，后来Yakovenko，D.Novikov，Horosov，Gavrilov，Petrov，Khovanski，Varchenko等人研究了这个问题。Glutsuk和作者在哈密顿多项式取一种特殊类型和任意次的情况下，对该问题的限制形式取得了一些进展。这个项目的目标之一是得到一个明确的上限估计，如果被积函数的次数较小，则该估计应该是哈密尔顿函数的次数的多项式的指数。动力系统理论一方面是决定论的领域，另一方面是混沌的领域。高维相空间中的矢量场形成了混沌领域。这一点在1960年的S就被理解了，从那时起，这个领域就成为数学家、计算机科学家和物理学家最感兴趣的课题。另一方面，从庞加莱和希尔伯特开始的一百多年来，被称为序域的平面微分方程这一经典学科引起了研究者的兴趣。希尔伯特的第16个问题是这一领域的主要问题。它坚持了数学家们100年来的努力，现在很明显，这个问题的简化的“受限”版本应该首先受到攻击。该项目基于新的想法和之前NSF支持的进展提出了一些具体的攻击方法。

The Hilbert 16th problem, part 2, is: "What may be said about the number and location of limit cycles of a planar polynomial vector field?" Traditionally this question is interpreted as a problem of finding an upper bound of the number of limit cycles of a polynomial vector field as a function of the degree of the polynomials. Even for the degree two, the upper bound is not yet found, and its existence is not yet proved. It makes sense to consider "restricted versions" of the Hilbert 16th problem. Namely, the set or all polynomial vector fields is replaced by a particular subset of by a similar class. The examples are Lienard and Abel equations (the latter ones are polynomial in the phase variable with coefficients 1-periodic in time). Even for these equations the problem of the number of limit cycles stays open. It is solved by the PI with an extra restriction: Abel equations, and Lienard ones with the polynomial of odd degree, are considered together with an upper bound for the magnitudes of the coefficients; the upper estimate on the number of limit cycles depends on this upper bound of the magnitudes. This is the result from the prior NSF support. In the current project we try to get rid of this latter restriction, and to give an estimate of the number of limit cycles that depends on the degrees of the polynomials in the right hand side only; for Abel equations the coefficients should be trigonometric polynomials of given degree. We hope to use mighty tools of the theory of complex analytic foliations, growth and zeros theorems for holomorphic functions and methods developed under the prior NSF support. Another important problem to be studied is the infinitesimal Hilbert 16th problem. It requires to estimate the number of limit cycles generated by a small perturbation from the ovals of the Hamiltonian polynomial vector field; closed orbits of the latter field form continuous families. This problem is reduced to the estimate of the number of zeros of an Abelian integral, that is, an integral of a polynomial 1-form over the ovals of a real polynomial in the plane; the estimate should be given in terms of the degrees of the polynomial Hamiltonian function and of the integrand. This problem was investigated by the author since 69; later on by Yakovenko, D.Novikov, Horosov, Gavrilov, Petrov, Khovanski, Varchenko and others. Some progress was obtained by Glutsuk and the author for the restricted version of the problem when the Hamiltonian polynomial is taken of a special type and of arbitrary degree. One of the goals of this project is to get an explicit upper estimate that is expected to be an exponential of a polynomial of the degree of the Hamiltonian function, provided that the integrand has a smaller degree.The theory of dynamical systems is the realm of determinism, on one hand, and of chaos, on the other hand. Vector fields in the phase space of dimension higher than two form the realm of chaos. This was understood in 1960's, and henceforth, this realm is the subject of the top interest for mathematicians, computer scientists and physicists. On the other hand, the classical subject of planar differential equations which may be called "realmof order" attracted the interest of researchers during more that one hundred years, beginning with Poincare and Hilbert. Hilbert's 16th problem is the main one in this domain. It persists the efforts of mathematicians during 100 years, and it is clear now that simplified "restricted" versions of the problem should be attacked first. The project suggests some concrete ways of this attack based on new ideas and the progress from the prior NSF support.