Kamal Boussaf

Picard Values of p-Adic Meromorphic Functions ∗

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 +∞ 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 = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f2 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′ + af2 has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf3 and f′ + bf4 have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.

SETS OF RANGE UNIQUENESS IN

We study sets of range uniqueness (SRUs) for analytic functions inside a disc of an algebraically closed field

p

K

-ADIC FIELDS

complete with respect to an ultrametric absolute value. The SRUs we obtain are converging sequences. We first obtain results that look like those known in

Strictly analytic functions on

\mathbb{C}

p

but involve a weaker hypothesis than in

-adic analytic open sets

\mathbb{C}

New results on applications of Nevanlinna methods to value sharing problems

: let

INTERPOLATION FOR P-ADIC ENTIRE FUNCTIONS

(a_n)

P-adic sets of range uniqueness

be a sequence of limit