A SHORT NOTE ON A PAIR OF MEROMORPHIC FUNCTIONS IN A p-ADIC FIELD, SHARING A FEW SMALL ONES

Let f, g, be two meromorphic functions on a complete ultrametric algebraically closed field IK of characteristic 0, or two meromorphic functions in an open disk of IK, that are not quotients of bounded analytic functions by polynomials. If f and g share I.M. 9 small meromorphic functions then f = g. If f and g have finitely many poles and share I.M. 4 small meromorphic functions, then f = g. 1. Main results Let IK be a complete ultrametric algebraically closed field of characteristic 0. Let us fix a ∈ IK and let R ∈]0, +∞[. We denote by d(a, R −) the disk {x ∈ IK | |x − a| 0, (resp. r ∈]0, R[), let q(r) be such that |a q(r) | ≤ r, |a q(r)+1 | > r. We then denote by N (r, f) the counting function of the zeros of f , counting multiplicity, as usual: for all r > 0, we put N (r, f) = q(r) j=0 s j (log |(a J)| − log(r)). Moreover, we denote by N (r, f) the counting function of the poles of f , ignoring multiplicity as q(r) j=0 (log |(a J)| − log(r)). Next, we define the counting function of zeros of f as Z(r, f) = N (r, 1 f) and we put Z(r, f) = N (r, 1 f). Finally we put T (r, f) = max(Z(r, f), N (r, f)). Then