## Taiwanese Journal of Mathematics

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

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

#### 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

**Permanent link to this document**

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**

Primary: 32K05: Banach analytic spaces [See also 58Bxx]

Secondary: 94A17: Measures of information, entropy

**Keywords**

Hilbert's 13th problem superposition representation $\varepsilon$-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