Properties And Applications Of Nash Functions With Several Complex Variables

Holomorphic function is an important research object of function theory of several complex variables.Nash function is a special kind of algebraic holomorphic function,which is widely used.According to the properties of Nash functions and Nash mappings,this paper illustrates the relations between elementary functions and Bergman kernel functions of bounded symmetric fields and Nash functions,and lists the applications of the properties of Nash functions to common submanifolds.This paper is divided into four chapters.In chapter 1,based on the holomorphic function definition,the definitions of Nash function and Nash mapping and their equivalent forms are explained.In chapter 2,the properties of Nash functions,Nash maps and Nash sets are summarized and their proof methods are reviewed.Meanwhile,for some property theorems with existing conclusions but no proof processes,this chapter supplements their proof processes.In chapter 3,the relation between Nash function and elementary function and Bergman kernel function is established by using the properties of Chapter 2.For the elementary analytical functions in single complex variable,it is proved that the constant value function f(z)=c(c ∈ C)and general power function f(z)=zα z≠0,∞,z ∈ C,z ∈ Q)are Nash functions.In several complex variables,it is proved that the exponential function f(z)=ez(Z∈Ck)is not Nash function,the Bergman kernel function of unit sphere,the Bergman kernel function of unit multivariate column and the Bergman kernel function of four typical fields are Nash functions.At the same time,If the Bergman kernel functions of two bounded domains are Nash function,then the Bergman kernel function of their product domain is also Nash function.In chapter 4,we summarize the application of Nash function properties and explain how to apply Nash function properties to solve the existence of common submanifolds in several complex variables.