A stronger version of a question proposed by K. Mahler

TL;DR

This paper advances the understanding of transcendental functions that take algebraic values at algebraic points, extending previous results to broader subsets of the complex plane.

Contribution

The authors prove a stronger version of Mahler's question by constructing transcendental functions with algebraic values on larger subsets of a2a2, generalizing earlier results.

Findings

Established existence of transcendental functions with algebraic values on new subsets of a2a2.

Extended Mahler's question to broader contexts in complex analysis.

Provided constructive methods for such functions.

Abstract

In 1902, P. St\"ackel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in $\mathbb{C}$. Recently, the authors answered positively this question. In this paper, we prove a much stronger version of this result by considering other subsets of $\mathbb{C}$.