Borel exceptional values in several complex variables and their applications to shared values of shifts and difference operators

TL;DR

This paper extends classical value-sharing results to several complex variables, establishing new theorems on Borel exceptional values and applying them to difference operators and PDEs.

Contribution

It introduces the first systematic study of shared value problems for higher-order difference operators in several complex variables, using new Borel exceptional value results.

Findings

Established fundamental results on Borel exceptional values in several complex variables.

Derived conditions for transcendental entire functions satisfying difference equations.

Analyzed meromorphic solutions of partial differential-difference equations with growth estimates.

Abstract

In this paper, we investigate shared value problems for shifts and higher-order difference operators of meromorphic and entire functions in several complex variables. Using Nevanlinna theory in $\mathbb{C}^n$, we obtain new uniqueness theorems when functions share values counting or ignoring multiplicities, extending several classical one-variable results to higher dimensions. A key contribution of this work appears in Section 2, where we establish fundamental results on Borel exceptional values in several complex variables. These propositions provide the main tools for proving our principal theorems. As applications, we derive conditions ensuring that a transcendental entire function satisfies $\Delta_c^{k} \equiv d\hspace{.05cc} f$ and we study meromorphic solutions of certain partial differential-difference equations, obtaining growth estimates and structural descriptions of entire solutions. To the best of our knowledge, this is the first systematic study of such shared value problems for higher-order difference operators in several complex variables.