Solutions of certain Fermat-type partial differential-difference equations
Sujoy Majumder, Debabrata Pramanik
TL;DR
This work extends the study of Fermat-type functional equations to Fermat-type partial differential-difference equations in several complex variables, providing a complete classification of finite-order entire and meromorphic solutions for the equation $F^{m_1}(z)+f^{m_2}(z+c)=1$ in $\mathbb{C}^m$. Employing Nevanlinna theory in several variables and high-dimensional Clunie-type results for differential-difference polynomials, the authors identify two principal solution families: (i) $m_1=m_2=2$ yielding a sine-type solution with a linear phase and a $c$-periodic polynomial $P$, and (ii) $m_1=2$, $m_2=1$ yielding a quadratic form $f(z)=1-\left(-\frac{z_1+\cdots+z_m}{2m}+g(z)\right)^2$ with $g$ depending on coordinate differences and obeying a shift condition. The paper also extends these results to meromorphic solutions, including cases where derivatives share $\infty$ CM, and proves nonexistence results in several parameter regimes, thereby resolving an open problem posed by Xu and Wang in higher dimensions. Examples accompany the theory, and corollaries for specific settings (e.g., $c=0$) are provided to illustrate the classifications.
Abstract
The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial z_j}\right)^{m_1} + f^{m_2}(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m ) = 1,\] where $m_1$ and $m_2$ are positive integers such that $m_1+m_2>2$ and $(c_1, c_2, \ldots, c_m)\in \mathbb{C}^m$. The results of our paper generalize the result of Xu and Wang \cite {XW1} from $\mathbb{C}^2$ to $\mathbb{C}^m$. Also in the paper we give positive answer of the open problem addressed by Xu and Wang \cite {XW1}. Moreover plenty of examples are provided to illustrate our findings.