A Complete Characterization of Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in $\mathbb{C}^n$

Abstract

The primary objective of this paper is to determine the explicit existence form and structure of finite-order entire solutions in $\mathbb{C}^n$ of the following system of Fermat-type partial differential-difference equations: \[\begin{cases} \left(\frac{\partial f_1\left(z\right)}{\partial z_1}\right)^{n_1} + (f_2 \left(z+c\right)-f_1(z) )^{m_1}= 1, \medskip \left(\frac{\partial f_2\left(z\right)}{\partial z_1}\right)^{n_2} + (f_1 \left(z+c \right)-f_2(z) )^{m_2}= 1, \end{cases}\] for different choices of the positive integers $n_1$, $n_2$, $m_1$, and $m_2$, where $c=(c_1,c_2,\ldots,c_n)$. We characterize the precise structure of finite-order transcendental entire solutions and extend the results of Xu et al. \cite{XLL1} from the setting of $\mathbb{C}^2$ to the more general space $\mathbb{C}^m$. In addition, several examples are presented to demonstrate the effectiveness and sharpness of the main results.