Value distribution of p-adic meromorphic functions

Abstract

Let $K$ be an algebraically closed field of characteristic $0$, complete with respect to an ultrametric absolute value. Let $f$ be a transcendental meromorphic function in $K$. We prove that if all zeroes and poles are of order $\geq 2$, then $f$ has no Picard exceptional value different from zero. More generally, if all zeroes and poles are of order $\geq k\geq 3$, then $f^{(k-2)}$ has no exceptional value different from zero. Similarly, a result of this kind is obtained for the $k-th$ derivative when the zeroes of $f $ are at least of order $m$ and the poles of order $n$, such that $mn>m+n+kn$. If $f$ admits a sequence of zeroes $a_n$ such that the open disk containing $a_n$, of diameter $|a_n|$ contains no pole, then $f$ and all its derivatives assume each non-zero value infinitely often. Several corollaries apply to the Hayman conjecture in the non-solved cases. Similar results are obtained concerning ''unbounded '' meromorphic functions inside an ''open'' disk.