The arithmetic of entire functions under composition (Q677488)

Let \(f\) be a nonconstant entire function of \(N\) complex variables, and let \(g\) be a function holomorphic on the image of \(f\). Then \(h=g \circ f\) is entire, and we call \(f\) a ``right factor'' of \(h\). For entire functions \(f,g\) of \(N\) complex variables, we define \(f\leq g\) if \(f(z)= f(w)\) implies \(g(z)= g(w)\) for \(z,w\in \mathbb{C}^N\). This relation partially orders the entire functions of \(N\) complex variables. The paper under review shows, among other things, that any family of nonconstant entire functions of one complex variable has a ``greatest'' common right factor.