The Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

中文翻译：

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费马型方程亚纯解的限制

费马型泛函方程$(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$， 在哪里n和ķ是正整数，在复平面中被考虑。我们的重点是 (*) 形式的方程，其中不知道在以下四类函数中的一类或多类中是否存在非常量解：亚纯函数、有理函数、整函数、多项式。对于这样的方程，我们获得了 (*) 的超越解必须满足的 Nevanlinna 函数的估计，以及非常数有理解的类似估计。作为一个应用，它表明 (*) 的超越全解，当n=ķ(ķ− 1) 与ķ≥ 3，必须满足某个微分方程，这是已知结果的推广，当ķ= 3. 给出了 (*) 的全解和多项式解的已知不存在定理的替代证明。此外，还讨论了多项式解的次数的一些限制。