Value distribution of q-differences of meromorphic functions in several complex variables

TL;DR

This paper extends value distribution theory to several complex variables for q-difference operators, establishing key lemmas, theorems, and growth estimates for meromorphic solutions of q-difference equations.

Contribution

It introduces q-difference analogues of fundamental results in several complex variables' value distribution theory, including the second main theorem and Picard type theorems.

Findings

Established q-difference versions of the logarithmic derivative lemma.

Proved a second main theorem for hyperplanes and hypersurfaces in the q-difference setting.

Applied the theory to analyze growth of solutions to linear partial q-difference equations.

Abstract

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations.