Research On The Uniqueness Of Meromorphic Functions Of Several Complex Variables Concerning Difference Polynomials

The value distribution theory of meromorphic function has always been an important content in complex analysis theory.Many scholars have made outstanding achievements in the study of value distribution theory.Based on the classical and multivariable Nevanlinna theory of meromorphic function,this thesis extends the difference operator,defines a new class of difference polynomials P(f),and studies the uniqueness of meromoiphic functions f(z)and their difference polynomials P(f)in Cm.The detailed arrangements of this thesis are as follows.The first chapter is the introduction of this thesis,which introduces the research background,the research content and its arrangement.It also briefly introduces the classical and multivariable Nevanlinna value distribution theory of meromorphic function.The second chapter mainly studies the uniqueness theorems of multivariable meromorphic functions when they and their difference polynomials share small functions.Firstly,the definition of difference polynomial P(f)is given,which is the extension of the difference operator,and the special circumstances that ensures P(f)=Δηnf are showen.Then we give the uniqueness theorem for the multivariable meromorphic functions of finite order and their difference polynomials P(f)when they CM share two small functions and infinity.After that,we further study the case of substituting infinity for small function and give the corresponding result.The research results in this chapter can be regarded as the generalization of univariate meromorphic function to multivariable meromorphic function.The third chapter mainly studies the uniqueness theorems under super conditions.First,the uniqueness results of multivariable meromorphic functions f(z)satisfying condition and their difference polynomials P(f)are studied under the condition of 2CM and deficit.Then we study the uniqueness results of multivariable meromorphic functions f(z)and their difference polynomial P(f)under the condition of 2IM+1CM and deficit.The fourth chapter summarizes the research content and looks forward to the future research.