On the functional equation \(f^ n=e^{P_ 1}+\cdots + e^{P_ m}\) and rigidity theorems for holomorphic curves (Q1261826)

Let \(E_ N\) be the set of all functions \(e^{P_ 1}+\dots+e^{P_ m}\) where \(P_ 1,\dots,P_ m: \mathbb{C}\to\mathbb{C}\) are polynomials such that \(\deg P_ n\leq N\) (\(j=1,\dots,m\)). Referring to an earlier paper by \textit{J. F. Ritt} [Trans. Am. Math. Soc. 31, 654-659 (1929)] and by \textit{M. Green} [Trans. Am. Math. Soc. 195, 223-230 (1974; Zbl 0289.32016)], the author deals with the functional equation and the title. Theorem 1, in which \(f\) is assumed to be holomorphic on \(\{z\): \(\omega_ 1<\arg z<\omega_ 2\}\), \(\omega_ 2-\omega_ 1>\pi/N\), gives conditions under which the solution \(f\) is an element of \(E_ N\). Theorem 2 deals with the case \(m=4\). Two other results concerning holomorphic curves \(g:C\to \mathbb{P}_ 2\) and \(g:\mathbb{C} \to \mathbb{P}_ 3\), of more technical character, are also given.