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Article Dans Une Revue p-Adic Numbers, Ultrametric Analysis and Applications Année : 2014

p-adic differential polynomials when poles make a small function

Alain Escassut
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Weiran Lü
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Chung Chun Yang
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Résumé

Notation: We denote by A(IK) the IK-algebra of entire functions in IK, by M(IK) the field of meromorphic functions in IK, i.e. the field of fractions of A(IK) and by IK(x) the field of rational functions. Throughout the paper, a is a point in IK and R is a strictly positive number and we denote by d(a, R −) the "open" disk {x ∈ IK : |x − a| < R}, by A(d(a, R −)) the IK-algebra of analytic functions in d(a, R −) i.e. the IK-algebra of power series ∞ n=0 a n (x − a) n converging in d(a, R −) and we denote by M(d(a, R −)) the field of meromorphic functions inside 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-subalgebra of A(d(a, R −)) consisting of the bounded analytic functions in d(a, R −), i.e. which satisfy sup n∈IN |a n |R n < +∞. And we denote by M b (d(a, R −)) the field of fractions of A b (d(a, R −)). Finally, we denote by A u (d(a, R −)) the set of unbounded analytic functions in d(a, R −), i.e. A(d(a, R −)) \ A b (d(a, R −)). Similarly, we set M u (d(a, R −)) = M(d(a, R −)) \ M b (d(a, R −)). Let log be a real logarithm function of base b > 1 and we set log + (x) = max(0, log(x)). Let f ∈ M(IK) resp. f ∈ M(d(0, R −)) having no zero and no pole at 0. Let r ∈]0, +∞[ resp. r ∈]0, R[ and let γ ∈ d(0, r). If f has a zero of order n at γ, we put ω γ (h) = n. If f has a pole of order n at γ, we put ω γ (f) = −n and finally, if f (γ) = 0, ∞, we set ω γ (f) = 0. We must now recall the definition of the Nevanlinna functions in p-adic [1], [3], [4]. We denote by Z(r, f) the counting function of zeros of f in d(0, r), counting multiplicities, i.e. Z(r, f) = max(ω 0 , 0) log r + ω γ (f)>0, 0<|γ|≤r ω γ (f)(log r − log |γ|). Similarly, we denote by Z(r, f) the counting function of zeros of f in d(0, r), ignoring multiplicities, and set Z(r, f) = u log r + ω γ (f)>0, 0<|γ|≤r (log r − log |γ|) with u = 1 when ω 0 (f) > 0 and u = 0 else.
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hal-01920257 , version 1 (13-11-2018)

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Alain Escassut, Weiran Lü, Chung Chun Yang. p-adic differential polynomials when poles make a small function. p-Adic Numbers, Ultrametric Analysis and Applications, 2014. ⟨hal-01920257⟩
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