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

希尔伯特第16题，第二部分，是：“平面多项式向量场的极限环的数量和位置是什么？”传统上，这个问题被解释为多项式向量场的极限环数的上界作为多项式次的函数的问题。甚至对于二阶，它的上界也还没有找到，它的存在性也还没有证明。考虑希尔伯特第16问题的“限制版本”是有意义的。也就是说，集合或所有多项式向量场被替换为一个类似类的特定子集。例子是Lienard和Abel方程（后者是相位变量的多项式，在时间上是1周期的系数）。即使对于这些方程，极限环数的问题仍然是开的。它是用带有附加限制的PI来解决的：Abel方程和具有奇数次多项式的Lienard方程与系数大小的上界一起被考虑；极限环数的上限估计取决于这个幅度的上限。这是先前NSF支持的结果。在当前的项目中，我们试图摆脱后一种限制，并给出一个极限环数的估计，它只依赖于右边多项式的次数；对于阿贝尔方程，系数应为给定次的三角多项式。我们希望利用复解析叶理理论、全纯函数的增长定理和零定理以及在先前NSF支持下开发的方法的强大工具。另一个需要研究的重要问题是无穷小的Hilbert 16问题。它要求估计由哈密顿多项式向量场的椭圆引起的小扰动所产生的极限环的数目；后一场的闭合轨道形成连续族。这个问题被简化为一个阿贝尔积分的零数的估计，也就是说，一个多项式1型在平面上实多项式的椭圆上的积分；估计应该用多项式哈密顿函数和被积函数的次数来表示。作者从1969年开始研究这个问题；后来又有雅科文科、诺维科夫、霍洛索夫、加夫里洛夫、彼得罗夫、霍万斯基、瓦琴科等人。作者和Glutsuk对于取任意次的特殊类型哈密顿多项式时问题的限制版本取得了一些进展。这个项目的目标之一是得到一个显式的上估计，该估计预计是哈密顿函数的次多项式的指数，前提是被积函数的次较小。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。二维以上相空间中的向量场形成混沌域。这在20世纪60年代就被理解了，从此以后，这个领域成为数学家、计算机科学家和物理学家最感兴趣的主题。另一方面，从庞加莱和希尔伯特开始，平面微分方程的经典课题——可称为“有序实域”——在一百多年的时间里吸引了研究人员的兴趣。希尔伯特的第16个问题是这个领域的主要问题。它坚持了数学家100年的努力，现在很清楚，应该首先攻击这个问题的简化“限制”版本。该项目提出了一些具体的方法，这些方法基于新的想法和先前NSF支持的进展。

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.