$ \mathbb{C}^n $中Fermat型偏微分差分方程的解
Solutions of Fermat-type partial differential-difference equations in $ \mathbb{C}^n $
摘要：
对于两个亚纯函数$ f $和$ g $，方程$ f^m+g^m=1 $可以看作是Fermat型方程.利用多复变亚纯函数的Nevanlinna理论，研究了$ \mathbb{C}^n $中Fermat型差分方程和偏微分-差分方程的超越整解的性质.此外，我们还得到了有限阶Fermat型偏微分差分方程$$\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^2+(f(z_1+c_1,z_2+c_2)-f(z_1,z_2))^2=1$$和$$f^2(z_1,z_2)+P^2(z_1,z_2)\left(\frac{\partial f(z_1+c_1,z_2+c_2)}{\partial z_1}-\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^2=1,$$在$ \mathbb{C}^2 $中的超越整体解的精确形式，其中$P(z_1,z_2)$是$\mathbb{C}^2$中的多项式.此外，本文的一个主要结果显著改进了Xu和Cao [Mediterr.（2018）15：227，1-14和Mediterr.（2020）17：8，1-4]。
英文摘要：
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].
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