## Taiwanese Journal of Mathematics

### A VERSION OF HILBERT’S 13TH PROBLEM FOR ENTIRE FUNCTIONS

Shigeo Akashi

#### Abstract

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 fuctions 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.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 6 (2008), 1335-1345.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405029

Digital Object Identifier
doi:10.11650/twjm/1500405029

Mathematical Reviews number (MathSciNet)
MR2444861

Zentralblatt MATH identifier
1157.32002

Subjects
Secondary: 94A17: Measures of information, entropy

#### Citation

Akashi, Shigeo. A VERSION OF HILBERT’S 13TH PROBLEM FOR ENTIRE FUNCTIONS. Taiwanese J. Math. 12 (2008), no. 6, 1335--1345. doi:10.11650/twjm/1500405029. https://projecteuclid.org/euclid.twjm/1500405029

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