The Arithmetic of Entire Functions under Composition

Abstract

In this paper, we prove, among other things, that any family of nonconstant entire functions of one complex variable has a greatest common right factor under composition. We prove a corresponding result for any family of pairwise dependent entire functions of N complex variables. Since f and af +b, where a, b # C and a{0 have all the same properties from the point of view of factoring under composition, we once and for all identify them, but continue (perhaps a little improperly) to talk about the equivalence classes as ``functions.'' Moreover we identify f and f b A where A is a biholomorphic one-to-one map of C onto C. Here are some definitions. Let f = f (z1 , ..., zn) be a nonconstant entire function of N complex variables, and let g= g(w) be a nonconstant holomorphic function in the region consisting of the complex plane minus any possible value omitted by f. Then the composition h=g b f, h(z1 , ..., zN)=g( f (z1 , ..., zN)) is a welldefined entire function, and in this case we call f a right factor of h. If we have a family of the form [h:]=[ g: b f ], then we say that f is common right factor of [h:]. If we have h=g: b f: , where : runs over an index set A, then we call h a common left multiple of the f: . For entire functions f and g of N complex variables, define f g if f (z)= f (w) implies g(z)=g(w), z, w # C. Then we remark that f g if and only if f is a right factor of g. The ``if '' part is trivial. For the ``only article no. 0087