Algebraic Values of Transcendental Functions at Algebraic Points

TL;DR

This paper proves that for any subset of algebraic numbers, there exists a transcendental entire function with that set as its exceptional set, and extends this result to a broader context with a unified proof.

Contribution

It introduces a generalization of the theorem on exceptional sets of transcendental functions and provides a unified proof approach.

Findings

Any subset of algebraic numbers can be the exceptional set of some transcendental entire function

The theorem is generalized to a more comprehensive setting

A unified proof method is presented

Abstract

In this paper, the authors will prove that any subset of $\overline{\QQ}$ can be the exceptional set of some transcendental entire function. Furthermore, we could generalize this theorem to a much more general version and present a unified proof.