ANDERSÉN–LEMPERT-THEORY WITH PARAMETERS: A REPRESENTATION THEORETIC POINT OF VIEW

2005 ◽  
Vol 04 (03) ◽  
pp. 325-340 ◽  
Author(s):  
FRANK KUTZSCHEBAUCH
Keyword(s):  
Vector Space ◽  
Vector Fields ◽  
Invariant Subspaces ◽  
Linear Representation ◽  
Complex Variables ◽  
Point Of View ◽  
Several Complex Variables ◽  
Algebraic Vector ◽  
Theoretic Point ◽  
Parametric Version

We calculate the invariant subspaces in the linear representation of the group of algebraic automorphisms of ℂnon the vector space of algebraic vector fields on ℂnand more generally we do this in a setting with parameter. As an application to the field of Several Complex Variables we get a new proof of the Andersén–Lempert observation and a parametric version of the Andersén–Lempert theorem. Further applications to the question of embeddings of ℂkinto ℂnare announced.

On the "Edge of the Wedge" Theorem

1963 ◽  
Vol 15 ◽  
pp. 125-131 ◽  
Author(s):  
Felix E. Browder
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Complex Variables ◽  
Point Of View ◽  
Several Complex Variables ◽  
Quantum Field ◽  
Complete Proof ◽  
Strategic Role ◽  
Mathematical Justification ◽  
Axiomatic Quantum Field Theory

In the mathematical justification of the formal calculations of axiomatic quantum field theory and the theory of dispersion relations, a strategic role is played by a theorem on analytic functions of several complex variables which has been given the euphonious name of the edge of the wedge theorem. The statement of the theorem seems to be due originally to N. Bogoliubov (cf. 3, Mathematical Appendix, pp. 654-673) but no complete proof which is fully satisfactory from the mathematical point of view has yet appeared in the literature.

scholarly journals On groups of formal diffeomorphisms of several complex variables

2012 ◽  
Vol 84 (4) ◽  
pp. 873-880
Author(s):  
Mitchael Martelo ◽  
Bruno Scárdua
Keyword(s):  
Vector Fields ◽  
Holonomy Group ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Holomorphic Foliations ◽  
Starting Point ◽  
Arbitrary Codimension ◽  
Dimension One ◽  
Phd Thesis ◽  
Integrating Factors

In this note we announce some results in the study of groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the holonomy group notion of a foliation's leaf. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, among others). Such system of equivalences also characterizes the existence of suitable integrating factors, i.e., invariant vector fields and one-forms associated to the group. Our aim is to state the basic lines of such dictionary for the case of several complex variables groups. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimension. The results in this note are derived from the PhD thesis "Grupos de germes de difeomorfismos complexos em várias variáveis e formas diferenciais" of the first named author (Martelo 2010).

scholarly journals A study on bornological properties of the space of entire functions of several complex variables

2002 ◽  
Vol 33 (4) ◽  
pp. 289-302 ◽  
Author(s):  
Mushtaq Shaker Abdul-Hussein ◽  
G. S. Srivastava
Keyword(s):  
Power Series ◽  
Vector Space ◽  
Entire Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Locally Convex Spaces ◽  
Classical Analysis ◽  
Analysis Theory ◽  
Important Position ◽  
Locally Convex

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$ and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

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.

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

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