Linear forms in algebraic points of Abelian functions. II

1976 ◽  
Vol 79 (1) ◽  
pp. 55-70 ◽  
Author(s):  
D. W. Masser
Keyword(s):  
Analytic Function ◽  
Maximum Modulus ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Maximum Modulus Principle ◽  
Linear Forms ◽  
Extrapolation Procedure ◽  
Abelian Functions

In this paper we continue to develop the apparatus needed for the proof of the theorem announced in (11). We retain the notation of (11) together with the assumptions made there about the field of Abelian functions. This section deals with properties of more general functions holomorphic on Cn. When n = 1 the extrapolation procedure in problems of transcendence is essentially the maximum modulus principle together with the act of dividing out zeros of an analytic function. For n > 1, however, this approach is not possible, and some mild theory of several complex variables is required. This was first used in the context of transcendence by Bombieri and Lang in (2) and (12), and we now give a brief account of the basic constructions of their papers.

Maximum modulus algebras and singularity sets

1980 ◽  
Vol 86 (3-4) ◽  
pp. 327-331 ◽  
Author(s):  
John Wermer
Keyword(s):  
Analytic Function ◽  
Maximum Principle ◽  
Maximum Modulus ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Ideal Space ◽  
Analytic Structure ◽  
Analogous Condition ◽  
Classical Theorem ◽  
Analytic Variety

SynopsisA classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.

scholarly journals Functional Dependence and Analytic Functions

1979 ◽  
Vol 22 (3) ◽  
pp. 367-369
Author(s):  
P. Ramankutty
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Function Theory ◽  
Functional Dependence ◽  
Complex Variables ◽  
Several Complex Variables ◽  
The Real ◽  
Cauchy Theorem ◽  
Complex Valued ◽  
Constant Complex

AbstractWithout appealing to the Cauchy theorem or its corollaries, it is proved that the real and imaginary parts of a non-constant complex-valued analytic function of several complex variables are functionally independent. This unifies and generalizes some results sporadically treated in standard treatises on function theory.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

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