Meromorphic functions of uniqueness
Résumé
Let $E$ be an algebraically closed field of characteristic $0$ which is either $\C$ or a complete ultrametric field $K$. We consider the composition of meromorphic functions $h\circ f$ 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 $h\circ f(a_m)=h\circ g(a_m)$ where $(a_m)$ 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 $(c_n)$ of $h'$ and the values $h(c_n)$ 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 $n\geq 3$ 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.
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