Exceptional values of p-adic analytic functions and derivatives

Abstract

Let 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 ∈ K, then c = 0. The space of the entire functions solutions of the equation y″ = φy, with φ 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. If f is meromorphic with finitely many zeroes then for every c ≠ 0, f′ − c has an infinity of zeroes. If 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 K has residue characteristic zero, we extend some results for entire functions to analytic functions in D.