Question

Let F, G be meromorphic functions such that Fⁿ+Gⁿ=1, assuming n≥4 if necessary. Prove that F and G are constants.

Answer 1

Let us first consider a function $h:z\mapsto (F(z),G(z))$ this function takes its values in the complex curve \mathcal{C}\subseteq \bar\mathbb{C}^2 defined by an equation $z^4+w^4=1$. This curve is a compact complex curve of genus $\frac{1}{2} (n-1)(n-2)$ which is $\geq 2$ for $n\geq 4$. By the Riemann uniformization theorem, a compact complex curves of genus $\geq 2$ admit a universal covering by a unit disc $D$. Therefore, there exists a lifting $\bar{h}: \mathbb{C}\to D$ and by the Liouville's theorem it is constant (as it is a bounded entire function). As $h=\pi\circ \bar h$, where $\pi$ is the covering map of the curve $\mathcal{C}$, we have $h=const$ and thus $F(z),G(z)=const$.

The condition $n\geq 4$ is not only sufficient, but necessary - for $n=1$ there are obvious linear solutions, for $n=2$ we can take trigonometric functions and for $n=3$ there are non-trivial solutions using the elliptic functions.