The factorization of $A(z)+B(w)$ under composition
Lee A. Rubel, Chung-Chun Yang · Illinois Journal of Mathematics · 1995
IntroductionTHEOREM.Let A and B be non-constant entire functions of one complex variable and suppose that f(g(z,w)) =A(z) + B(w) for all z, w C, where f is an entire function of one variable and g is an entire function of two variables.Then f must be affine: f(s r) ag" +/3, a,/3 C and g must have the form g(z,w) a(z) + b(w)for some entire functions a(z) and b(w) of one variable.This theorem says that the only entire factorizations (under composition of functions) of A(z) + B(w) are the obvious ones.Note that the theorem is a global result dealing with entire functions.There is no corresponding local result--witness A( z) + B(w) log(exp(A(z) + B(w)),where and -1 are analytic functions suitably defined on regions in C.Besides some algebraic and analytic manipulating of an elementary kind, the main tool in the proof is Nevanlinna theory based on exhaustions of C2 by polydiscs.In particular, we use a version of the lemma of the logarithmic