Resumen de Meromorphic functions of uniqueness

Alain Escassut

• Let E be an algebraically closed field of characteristic 0 which is either or a complete ultrametric field K. We consider the composition of meromorphic functions h?f where h is meromorphic in all E and f is meromorphic either in E or in an open disk of K. We then look for a condition on h in order that if 2 similar functions f,g satisfy h?f(am)=h?g(am) where (am) is a bounded sequence satisfying certain condition, this implies f=g. Particularly we generalize to meromorphic functions previous results on polynomials of uniqueness. The condition on h involves the zeros (cn) of h' and the values h(cn) but is weaker than this introduced by H. Fujimoto (injectivity on the set of zeros of h'). The main tool is the Nevanlinna Theory but also involves some specific p-adic properties and basic affine properties. Results concerning p-adic entire functions only suppose a property involving 2 zeros of h'. Polynomials of uniqueness for entire functions are characterized. Every polynomial P of prime degree n3 is a polynomial of uniqueness for p-adic entire functions, except if is of the form A(x-a)n+B. A polynomial P such that P' has exactly two distinct zeros is a polynomial of uniqueness for meromorphic functions in K if and only if both zeros have a multiplicity order greater than 1. Results on p-adic functions have applications to rational functions in any field of characteristic 0.