Fermat型函数方程解的存在性

The Existence of Fermat-type Functional Equations

首先给出了费马型函数方程f6(z)+g6(z)+h6(z)=1的一类非常数整函数解存在的必要条件;其次,证明了当f2(z)+g2(z)+h2(z)=0时,函数方程f6(z)+g6(z)+h6(z)=3没有非常数的整函数解;最后得到函数方程f8(z)+g8(z)+h8(z)=z没有级小于18的亚纯函数解的结论。

This paper first presents a necessary condition for the existence of nonconstant entire functional solutions with a class of the Fermat functional equations f6（z）＋g6（z）＋h6（z）=1; Secondly, it has been proved that f2（z）＋g2（z）＋h2（z）=0, there is no nonconstant entire functional solutions satisfying the functional equation f6（z）＋g6（z）＋h6（z）=3 ;Finally it can be concluded that nonconstant meromorphic functional solutions with order smaller 1 than satisfying the equation does not exsit f8（z）＋g8（z）＋h8（z）=z.