ZEROS OF ULTRAMETRIC MEROMORPHIC FUNCTIONS f′ fn(f − a)k− α

Let 𝕂 be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. Similarly to the Hayman problem, here we study meromorphic functions in 𝕂 or in an open disk that are of the form f′ fn(f − a)k− α with α a small function, in order to find sufficient conditions on n, k assuring that they have infinitely many zeros. We first define and characterize a special value for a meromorphic function and check that, if it exists, it is unique. So, such values generalize Picard exceptional values.