| Abstract: | The functional equation \(f^{m}+g^{m}=1\) can be regarded as the Fermat-type equations over function fields. In this paper, we investigate the entire and meromorphic solutions of the Fermat-type functional equations such as partial differential-difference equation \(\left( \frac{\partial f(z_{1}, z_{2})}{\partial z_{1}}\right) ^{n}+f^{m}(z_{1}+c_{1}, z_{2}+c_{2})=1\) in \(\mathbb {C}^{2}\) and partial difference equation \(f^{m}(z_{1}, \ldots , z_{n})+f^{m}(z_{1}+c_{1}, \ldots , z_{n}+c_{n})=1\) in \(\mathbb {C}^{n}\) by making use of Nevanlinna theory for meromorphic functions in several complex variables. |
