On generalized Fermat Diophantine functional equations in \(\mathbf{C}^n\) and Picard type theorems. (English) Zbl 07861425
Summary: This paper concerns entire and meromorphic solutions of the generalized Fermat Diophantine functional equations \(hf^p + kg^q = 1\) in \(\mathbf{C}^n\), where \(h,k\) are meromorphic coefficients in several complex variables and \(p,q \geq 1\) are integers with \((p,q)\neq (1,1)\). As applications, we determine when entire solutions of the simple-looking functional equation \(f^p + g^q = 1\) in \(\mathbf{C}\) reduce to constant and then apply the result to show two well-known Picard type theorems in a direct manner.
MSC:
| 32A15 | Entire functions of several complex variables |
| 32A20 | Meromorphic functions of several complex variables |
| 32A22 | Nevanlinna theory; growth estimates; other inequalities of several complex variables |
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