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

Abstract : 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| < R}. We denote by A(IK) the IK-algebra of entire functions in IK and by M(IK) the field of meromorphic functions which is its field of fractions. We denote by A(d(a, R −)) the IK-algebra of analytic functions in d(a, R −) i.e. the set of power series converging in the disk d(a, R −). and by M(d(a, R −)) the field of mero-morphic functions in d(a, R −) i.e. the field of fractions of A(d(a, R −)). Moreover , we denote by A b (d(a, R −)) the IK-algebra of functions f ∈ A(d(a, R −)) that are bounded in d(a, R −), by M b ((. a, R −)) its field of fractions and we put M u (d(a, R −)) = M((. a, R −)) \ A b (. a, R −)) We define N (r, f) ([1], chapter 40 or [3], chapter 2) in the same way as for complex meromorphic functions [2]. Let f be a meromorphic function in all IK having no zero and no pole at 0. Let (a n) n∈IN be the sequence of poles of f , of respective order s n , with |a n | ≤ |a n+1 | and, given r > 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 0
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Alain Escassut, C. Yang. A SHORT NOTE ON A PAIR OF MEROMORPHIC FUNCTIONS IN A p-ADIC FIELD, SHARING A FEW SMALL ONES. 2019. ⟨hal-02377422⟩

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