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

Alain Escassut (1) , C C Yang

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

A new Nevanlinna theorem on q p-adic small functions is given. 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 7 small meromorphic functions I.M., then f = g. Better results hold when f and g satisfy some property of growth. Particularly , if f and g have finitely many poles or finitely many zeros and share 3 small meromorphic functions I.M., 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 meromorphic 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 (d(a, R −)) its field of fractions and we put M u (d(a, R −)) = M(d(a, R −)) \ A b (d(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 (resp. in d(0, R −)) 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 0

Mots clés: p-adic meromorphic functions; small functions; sharing small functions 1991 Mathematics Subject Classification Primary 12J25; Secondary 30D35; 30G06

Domaines: Théorie des nombres [math.NT]

Alain Escassut, C C Yang. A SHORT NOTE ON A PAIR OF MEROMORPHIC FUNCTIONS IN A p-ADIC FIELD, SHARING A FEW SMALL ONES. Rendiconti del Circolo Matematico di Palermo, In press. ⟨hal-02568414⟩