On generalized Fermat Diophantine functional and partial differential equations in $\mathbf{C}^2$

TL;DR

This paper characterizes meromorphic solutions to generalized Fermat Diophantine functional equations and related partial differential equations in two complex variables, extending understanding of their structure and solutions.

Contribution

It provides a comprehensive characterization of solutions to these equations in ^m + k g^n = 1 in ^2, including solutions to associated PDEs, for the first time.

Findings

Explicit forms of solutions are derived.

Conditions for existence of solutions are established.

Connections between functional and differential equations are explored.

Abstract

In this paper, we characterize meromorphic solutions $f(z_1,z_2),g(z_1,z_2)$ to the generalized Fermat Diophantine functional equations $h(z_1,z_2)f^m+k(z_1,z_2)g^n=1$ in $\mathbf{C}^2$ for integers $m,n\geq2$ and nonzero meromorphic functions $h(z_1,z_2),k(z_1,z_2)$ in $\mathbf{C}^2$. Meromorphic solutions to associated partial differential equations are also studied.