Picard values of $p$-adic meromorphic functions
Résumé
We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field $K$, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman's alternative theorem both in $K$ and in any open disk. Here the main hypothesis is based on the behaviour of $|f|(r)$ when $r$ tends to $+\infty$ on properties of special values and quasi-exceptional values. We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman's conjectures for $n=1$ and for $n=2$. Given a meromorphic transcendental function $f$, at least one of the two functions $f'f$ and $f'f^2$ assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on $K$ admits no infinite stationary subsequence, then either $f'+af^2$ has infinitely many zeros that are not zeros of $f$ for every $a\in K^*$ or both $f'+bf^3$ and $f'+bf^4$ have infinitely many zeros that are not zeros of $f$ for all $b\in K^*$. Most of results have a similar version for unbounded meromorphic functions inside an open disk.