Solutions of Fermat-type partial differential-difference equations in $ \mathbb{C}^n $

TL;DR

This paper investigates the properties and explicit forms of transcendental entire solutions to Fermat-type partial differential-difference equations in several complex variables, using Nevanlinna theory, and improves previous results in the field.

Contribution

It provides a detailed analysis of solutions to Fermat-type equations in multiple complex variables and determines their precise form in specific cases, advancing the understanding of such equations.

Findings

Characterization of transcendental entire solutions in $\mathbb{C}^2$

Explicit form of solutions for specific equations

Improved results over previous studies by Xu and Cao

Abstract

For two meromorphic functions $ f $ and $ g $, the equation $ f^m+g^m=1 $ can be regarded as Fermat-type equations. Using Nevanlinna theory for meromorphic functions in several complex variables, the main purpose of this paper is to investigate the properties of the transcendental entire solutions of Fermat-type difference and partial differential-difference equations in $ \mathbb{C}^n $. In addition, we find the precise form of the transcendental entire solutions in $ \mathbb{C}^2 $ with finite order of the Fermat-type partial differential-difference equation $$\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^2+(f(z_1+c_1,z_2+c_2)-f(z_1,z_2))^2=1$$ and $$f^2(z_1,z_2)+P^2(z_1,z_2)\left(\frac{\partial f(z_1+c_1,z_2+c_2)}{\partial z_1}-\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^2=1,$$ where $P(z_1,z_2)$ is a polynomial in $\mathbb{C}^2$. Moreover, one of the main results of the paper significantly improved the result of Xu and Cao [Mediterr. J. Math. (2018) 15:227 , 1-14 and Mediterr. J. Math. (2020) 17:8, 1-4].