The functional equation P(f) = Q(g) in a p-adic field

Let K be a complete ultrametric algebraically closed field of characteristic pi. Let P, Q be in K[x] with P'Q' not identically 0. Consider two different functions f, g analytic or meromorphic inside a disk \x - a\ <r (resp. in all K), satisfying P(f) = Q(g). By applying the Nevanlinna's values distribution Theory in characteristic pi, we give sufficient conditions on the zeros of P, a to assure that both f, g are "bounded" in the disk (resp. are constant). If pi not equal 2 and deg(P) = 4, we examine the particular case when Q = lambdaP (lambda is an element of K) and we derive several sets of conditions characterizing the existence of two distinct functions f, g meromorphic in K such that P(f) =lambdaP(g). (C) 2003 Elsevier Inc. All rights reserved.