Exceptional values of $p$-adic derivatives \ A survey with some improvements

Abstract

Let $\K$ be a complete ultrametric algebraically closed field of characteristic $0$ and let $f$ be a meromorphic function in $\K$ admitting primitives. We show that $f$ has no value taken finitely many times provided an additional hypothesis is satisfied: either $f$ has finitely many poles of order $\geq 3$, or $f$ has two perfectly branched values, or the logarithm of the number of poles in the disk of center $0$ and diameter $r$ is bounded by $O(\Log(r))$ ($r>1$). We make the conjecture: all additional hypotheses are superfluous.