Real meromorphic functions

摘要本文拟继续运用前人所发展的几何方法研究实亚纯函数的根和临界点的分布。本文拟进行的主要研究方向如下：a)实域和复域Wronski图及其相关极点赋值图的研究。b)实整函数的连续导数的根分布的研究。c)实函数的振荡率与其谱性质之间关系的进一步研究。d)紧曲面上具有二次奇异点的正曲率测度的存在唯一性研究。数学及其应用中的一个基本问题是一个给定的方程或方程组是否有解，有多少解，以及它们在哪里。在亚纯函数理论中，研究了一类方程f(z)=a的这些问题，其中a是给定复数，f是给定亚纯函数。亚纯函数类包括初等函数，如有理函数、指数函数和三角函数，也包括特殊函数，即高等超越函数，如伽玛函数、艾里函数、椭圆函数等。在数学应用中出现的大多数函数都属于这一类。在现代数学中，关于方程可解性的问题通常用几何语言表述，这使得结果对我们的几何直觉很有吸引力。拟继续研究亚纯函数的几何理论。大多数提出的研究都与实解的存在性有关，这是一个比通常研究的复解的存在性更微妙的问题。最初的动机之一（除了这些问题内在的数学重要性）是所谓的“极点放置问题”，这是线性系统控制理论中一个主要的未解决的数学问题。这一领域的研究结果将对复杂自动控制系统的设计产生影响。这些结果将限制用某种类型的控制装置来控制一个给定大小的系统的可能性。更广泛影响的另一个重要领域是最近发现的这一提议中所考虑的问题与物理学的联系，更准确地说，与铁磁性的精确可解模型的联系。

ABSTRACTThe proposer intends to continue his study of the distributionof roots and critical points of real meromorphic functionsusing the geometric methods developed in his previous work.The main directions of the proposed research are the following.a) The study of the Wronski map, both in the real and complex domains, and of the related pole assignment map.b) The study of the distribution of roots of successivederivatives of real entire functions.c) Further investigation of the relation between the rateof oscillation of real functions and their spectral properties.d) The study of existence and uniqueness ofmetrics of positive curvature with conic singularities on compact surfaces.One of the basic questions in mathematics and its applications iswhether a given equation or a system of equations has solutions,how many, and where are they located. In the theory of meromorphic functions one studies these questions for equations of the typef(z)=a, where a is a given complex number and f a given meromorphic function. The class of meromorphic functions includes elementary functions, such as rational, exponential and trigonometric ones, as well as the special functions, a. k. a. higher transcendental functions, such as the Gamma function, Airy functions, elliptic functions and so on. Most functions arising in applications of mathematics belong to this class. In modern mathematics, questions about resolvability of equations are usually formulated in geometric language, which makes the results appealing to our geometric intuition.The proposer plans to continue his study of geometric theory of meromorphic functions. Most of the proposed research is related to existence of real solutions, a moresubtle question than the existence of complex solutions, which are usually studied. One of the original motivations (beside intrinsic mathematical importance of these questions) was the so-called "pole placement problem", which is a major unsolved mathematical problem in control theory of linear systems. The results in this area will have implications for the design of complicated automatic control systems. These results would establish limitations on the possibility to control a system of given size by a control device of certain class. Another important area of the broader impact is the recently discovered connection of the problems considered in this proposal with physics, more precisely, with the exactly solvable models of ferromagnetism.