Exceptional values of p-adic analytic functions and derivatives

Abstract

International audienceLet $K$ be an algebraically closed field of characteristic $0$, complete with respect to an ultrametric absolute value $|\ . \ |$. Given a meromorphic function $f$ in $K$ (resp. inside an ''open'' disk $D$) we check that the field of small meromorphic functions in $K$ (resp. inside $D$) is algebraically closed in the whole field of meromorphic functions in $K$ (resp. inside $D$). If two analytic functions $h,\ l$ in $K$, other than affine functions, satisfy $h'l-hl'=c\in K$, then $c=0$. The space of the entire functions solutions of the equation $y''=\phi y$, with $\phi$ a meromorphic function in $K$ or an unbounded meromorphic function in $D$, is at most of dimension 1. If a meromorphic function in $K$ has no multiple pole, then $f'$ has no exceptional value. Let $f$ be a meromorphic function having finitely many zeroes. Then for every $c\neq 0$, $f'-c$ has an infinity of zeroes. If ${1\over f}$ is not a constant or an affine function and if $f$ has no simple pole with a residue equal to $1$, then $f'+f^2$ admits at least one zero. When the field $K$ has residue characteristic zero, then we can extend to analytic functions in $D$ some results showed for entire functions