Rigidity of entire functions sharing a finite set with their partial derivatives in C^n

TL;DR

This paper extends the study of entire functions sharing a finite set with their derivatives from one complex variable to several complex variables, using normality criteria to derive existence conditions.

Contribution

It generalizes previous single-variable results to multiple complex variables, broadening the understanding of such functions in higher dimensions.

Findings

Derived necessary conditions for the existence of such functions in C^n.

Extended earlier single-variable results to several complex variables.

Provided a comprehensive framework for the rigidity of these functions.

Abstract

This paper investigates certain classes of entire functions in C^n that, together with their partial derivatives, share a finite set consisting of three elements. By employing normality criteria, we study the behaviour of such functions and derive the necessary conditions governing their existence. Our results extend those of [4], originally established for functions of a single complex variable, to the setting of several complex variables, thereby providing a comprehensive generalization of the earlier result in a direction not previously explored.