Exceptional values of p-adic derivatives A survey with some improvements - Escassut, Alain (Sarajevo Journal of Mathematics, 2024)
Let be a complete ultrametric algebraically closed field of characteristic 0 and let f be a meromorphic function in admitting primitives. We show that f has no value taken finitely many times provided an additional hypothesis is satisfied: either f has finitely many poles of order ≥ 3, or f has two perfectly branched values, or the logarithm of the number of poles in the disk of center 0 and diameter r is bounded by O((r)) (r>1). We make the conjecture: all additional hypotheses are superfluous.