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arXiv · 2606.05240

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

Sujoy Majumder·Jhilik Banerjee·Abhijit Banerjee·Elias G. Saleeby
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Abstract

The primary objective of this paper is to determine the existence and explicit form 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 several choices of the positive integers $n_1$, $n_2$, $m_1$, and $m_2$, where $c=(c_1,c_2,\ldots,c_n)$. We obtain structural classifications and nonexistence results in the exponent regimes specified in the main theorems, extending results of Xu et al. \cite{XLL1} from $\mathbb{C}^2$ to $\mathbb{C}^n$. Several examples illustrate the resulting solution families.

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Sujoy Majumder, Jhilik Banerjee, Abhijit Banerjee, Elias G. Saleeby. 2026-06-03. Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in $\mathbb{C}^n$. https://arxiv.org/abs/2606.05240

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