论文标题
$ \ mathbb {c}^n $中的fermat-type部分差分方程的解决方案
Solutions of Fermat-type partial differential-difference equations in $ \mathbb{C}^n $
论文作者
论文摘要
对于两个meromormorphic函数$ f $和$ g $，方程$ f^m+g^m = 1 $可以被视为fermat-type方程。在几个复杂变量中，使用Nevanlinna理论来实现meromormormormormorphic函数，本文的主要目的是研究Fermat-Type差异的先验整个解决方案的性质和$ \ Mathbb {C}^n $中的部分差异差异和部分差分差异方程。此外，我们在$ \ mathbb {c}^2 $中找到了整个解决方案的精确形式z_1} \ right）^2+（f（z_1+c_1，z_2+c_2）-f（z_1，z__2））^2 = 1 $$和$$ f^2（z_1，z_1，z_2）+p^2（z___1，z_1，z__2） z_1} - \ frac {\ partial f（z_1，z_2）} {\ partial z_1} \ right）^2 = 1，$ p（z_1，z_2）$是$ \ mthbb {c}^2 $的$ p（z_1，z_2）$。此外，本文的主要结果之一显着改善了Xu和Cao [Mediterr的结果。 J. Math。 （2018）15：227、1-14和Mediterr。 J. Math。 （2020）17：8，1-4]。
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].
