The factorization of \(A(z)+B(w)\) under composition (Q1892739)

All the functions involved are supposed to be entire and nonconstant. It is proved that the only ways \(A(z) + B(w)\) can be written as \(f(g(z,w))\) are the obvious ways, i.e. \(f\) must be linear: \(f(t) = at + b\). The proof uses Nevanlinna theory based on the exhaustion of \(\mathbb{C}^ 2\) by asymmetric polydiscs. It is to be emphasized that the restriction to entire functions is essential. No such result holds for functions that are just supposed holomorphic on proper open sets.