Some transcendental functions with an empty exceptional set

TL;DR

This paper constructs explicit examples of transcendental entire functions that have no algebraic points where the function takes algebraic values, thus having an empty exceptional set.

Contribution

It provides explicit examples of transcendental entire functions with an empty exceptional set, advancing understanding of the distribution of algebraic values.

Findings

Constructed explicit transcendental entire functions with empty exceptional set

Shows such functions can exist beyond classical examples

Contributes to the theory of algebraic values of transcendental functions

Abstract

A transcendental function usually returns transcendental values at algebraic points. The (algebraic) exceptions form the so-called \emph{exceptional set}, as for instance the unitary set $\{0\}$ for the function $f(z) = e^z \,$, according to the Hermite-Lindemann theorem. In this note, we give some explicit examples of transcendental entire functions whose exceptional set are empty.