Ludwig Bieberbach's "Remarks on Hilbert's Thirteenth Problem" and Addendum

TL;DR

This paper provides an English translation of Bieberbach's 1931 work on Hilbert's Thirteenth Problem, discussing conditions for functions of three variables to be constructed from two-variable functions, and addresses a critical error in the original proof.

Contribution

It offers the first English translation of Bieberbach's seminal paper and clarifies the conditions and limitations in representing multivariable functions through nested two-variable functions.

Findings

Identifies conditions for representability of three-variable functions

Acknowledges a critical error in Bieberbach's original proof

Connects Hilbert's Thirteenth Problem to differential equations

Abstract

This is an English translation of Ludwig Bieberbach's paper ``Remarks on Hilbert's Thirteenth Problem" originally written in German and originally published in Journal f\"ur die Reine und Angewandte Mathematik - 165 (89-92) 1931, along with an addendum to the paper published in 1933. Beiberbach studies under what conditions are there functions of three variables which can or cannot be obtained by combining or nesting functions of two variables. In the addendum, Bieberbach acknowledges a fatal error in a proof in the original article and connects Hilbert's thirteenth problem to a related problem in the theory of differential equations.