On Entire solutions of system of Fermat-type difference and partial differential-difference equations in $\mathbb{C}^n$

Abstract

The equation $f^n+g^n=1$, $n\in\mathbb{N}$ can be regarded as the Fermat Diophantine equation over the function field. In this paper we study the characterization of entire solutions of some system of Fermat type functional equations by taking $e^{g_1(z)}$ and $e^{g_2(z)}$ in the right hand side of each equation, where $g_1(z)$ and $g_2(z)$ are polynomials in $\mathbb{C}^n$. Our results extend and generalize some recent results. Moreover, some examples have been exhibited to show that our results are precise to some extent.