A positive answer for a question proposed by K. Mahler

TL;DR

This paper confirms the existence of uncountably many transcendental functions analytic in the complex plane that take algebraic values at algebraic points, extending Mahler's 1976 question.

Contribution

It provides a positive answer to Mahler's question by constructing uncountably many such functions, expanding the understanding of algebraic values of transcendental functions.

Findings

Existence of uncountably many such functions

Functions are transcendental and analytic in a9

These functions take algebraic values at all algebraic points

Abstract

In 1902, P. St\"{a}ckel 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 $\mathbb{C}$. In this work, we provide a positive answer for this question by showing the existence of uncountable many of these functions.