Finite Order Transcendental Entire Solutions of Coupled Fermat-Type Difference Equations in Several Complex Variables

Abstract

Motivated by recent developments in complex difference equations and Nevanlinna theory in several complex variables, we investigate finite-order transcendental entire solutions of the coupled Fermat-type difference system: \beas \begin{cases} f_1^{n_1}(z)+ f_2^{m_1} \left(z+c \right) = 1,\\ f_2^{n_2}(z) + f_1^{m_2} \left(z+c\right) = 1, \end{cases} \eeas where $z,c=(c_1,c_2,\dots ,c_n)\in {\mathbb{C}}^n$ for various choices of $n_i,m_i$, $i=1,2$. where $n_i,m_i\in \mathbb{N}$ and $n_i+m_i\ge 2$ $(i=1,2)$. Extending the classical investigations of Gross--Yang, Liu, Liu--Cao--Cao and more recently, Xu \emph{et al.} in one and two complex variables, to a general coupled system in ${\mathbb{C}}^n$ we establish a complete characterization of all finite-order transcendental entire solutions. We have determined that the solution structure is completely determined by the relative sizes of the exponents.

13 pages