New applications of the p-adic Nevanlinna Theory

Alain Escassut (1) , Ta Thi Hoai An

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis , here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in l C, we consider functions f n (x)f m (ax + b), g n (x)g m (ax + b) with |a| = 1 and n = m, sharing a rational function and we show that f g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n−m|∞ ≥ 5, then f n (x)f m (ax+b)−w has infinitely many zeros. Finally, we examine branched values for meromorphic functions f n (x)f m (ax + b).

Domaines: Théorie des nombres [math.NT]

Alain Escassut, Ta Thi Hoai An. New applications of the p-adic Nevanlinna Theory. p-Adic Numbers, Ultrametric Analysis and Applications, 2018. ⟨hal-01921858⟩