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A note on transcendental analytic functions with rational coefficients mapping $\mathbf{Q}$ into itself

Year: 2024

Type: article

Abstract: In this note, the main focus is on a question about transcendental entire functions mapping $\mathbf{Q}$ into $\mathbf{Q}$ (which is related to a Mahler's problem). In particular, we prove that, for any $t>0$, there is no a transcendental entire function $f\in\mathbf{Q}[[z]]$ such that $f(\mathbf{Q})\subseteq\mathbf{Q}$ and whose denominator of $f(p/q)$ is $O(q^{t})$, for all rational numbers $p/q$, with $q$ sufficiently large.

Source: Proceedings of the Japan Academy Series A Mathematical Sciences

DOI: 10.3792/pjaa.100.009

Authors: Jean Lelis, Diego Marques, Carlos Gustavo Moreira, Pavel Trojovský

Language: English

Topic: Meromorphic and Entire Functions

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

OpenAlex ID: W4403082356

FWCI:

Cites: 5

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Related to: 10