A VERSION OF HILBERT'S 13th PROBLEM FOR ENTIRE FUNCTIONS

It is famous that Hilbert proved that, for any positive integer n, there exists an entire function fn(·, ·, ·) of three complex variables which cannot be represented as any n-time nested superposition constructed from several entire frictions of two complex variables. In this paper, a finer classifi...

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Bibliographic Details
Published in:Taiwanese journal of mathematics 2008-09, Vol.12 (6), p.1335-1345
Main Authors: Akashi, Shigeo, 明石 重男
Format: Article
Language:English
Subjects:
Analytic functions
Complex variables
Continuous functions
Entire functions
Integers
Mathematical functions
Mathematical theorems
Mathematical variables
Numbers
Real variables
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Summary:It is famous that Hilbert proved that, for any positive integer n, there exists an entire function fn(·, ·, ·) of three complex variables which cannot be represented as any n-time nested superposition constructed from several entire frictions of two complex variables. In this paper, a finer classification of the 13th problem formulated by Hilbert is given. This classification is applied to the theorem showing that there exists an entire function f(·, ·, ·) of three complex variables which cannot be represented as any finite-time nested superposition constructed from several entire functions of two complex variables. The original result proved by Hilbert can be derived from this theorem.
ISSN:1027-5487
2224-6851
DOI:10.11650/twjm/1500405029