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

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: $$\{\begin{array}{l}{\left(\frac{\mathrm{\partial }{f}_1\left(z\right)}{\mathrm{\partial }{z}_1}\right)}^{n_1}+(f_2(z+c)-f_1(z){)}^{m_1}=1,\text{medskip}{\left(\frac{\mathrm{\partial }{f}_2\left(z\right)}{\mathrm{\partial }{z}_1}\right)}^{n_2}+(f_1(z+c)-f_2(z){)}^{m_2}=1,\end{array}$$ for different choices of the positive integers $n_1$, $n_2$, $m_1$, and $m_2$, where $c=(c_1,c_2,\dots ,c_n)$. 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.