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

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 $m,n\ge 2$ 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.