Diego Alves - On a version of the Problem B of Mahler involving derivatives

cm:12784 - Communications in Mathematics, April 3, 2024, Volume 32 (2024), Issue 1 - https://doi.org/10.46298/cm.12784
On a version of the Problem B of Mahler involving derivativesArticle

Authors: Diego Alves

    In 1902, Paul Stäckel constructed an analytic function $f(z)$ in a neighborhood of the origin, which was transcendental, and with the property that both $f(z)$ and its inverse, as well as its derivatives, assumed algebraic values at all algebraic points in this neighborhood. Inspired by this result, Mahler in 1976 questioned the existence of an transcendental entire function $f(z)$ such that $f(\overline{\mathbb{Q}})$ and $f^{-1}(\overline{\mathbb{Q}})$ are subsets of $\overline{\mathbb{Q}}.$ This problem was solved by Marques and Moreira in 2017. As Stäcklel's result involved derivatives, it is natural to question whether we have an analogous result for transcendental entire functions involving derivatives. In this article, we show that there are an uncountable amount of such functions.

    Volume: Volume 32 (2024), Issue 1
    Published on: April 3, 2024
    Accepted on: February 10, 2024
    Submitted on: January 5, 2024
    Keywords: Mathematics - Complex Variables,Mathematics - Number Theory

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