# Classical p-adic Nevanlinna theory and Nevalinna Theory out of a hole

Abstract : In the first section called Classical theory, we recall basic properties of the analytic and meromorphic functions, Motzkin's factoraization of analytic elements and the classical p-adic Nevanlinna theory. The second section is devoted to meromorphic functions in the complement of an "open" disk with the use of Motzkin's factorization and we show the existence of a Nevanlinna theory in that field of meromorphic functions. Applications are examined: Nevanlinna theory on 3 small functions, parametrization of algebraic curves, qusai-exceptional small functions and branched small functions. I. Classical theory I.1 Basic definitions We denote by IK an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value |. |. Analytic functions inside a disk or in the whole field IK were introduced and studied in many books. Given α ∈ IK and R ∈ IR * + , we denote by d(α, R) the disk {x ∈ IK | |x − α| ≤ R}, by d(α, R −) the disk {x ∈ IK | |x − α| < R} and by C(α, r) the circle {x ∈ IK | |x − α| = r}. We denote by |IK| the set {|x| | x ∈ IK}. During the forties, Mark Krasner had the idea of introducing a kind of analytic functions based upon the following property in complex analysis. Let D be an open bounded subset of l C. Then, by Runge's Theorem, every holomorphic functions in D which is continuous on the closure of D is the uniform limit on D of sequences of rational functions with no pole in D. This property suggested Marc Krasner defining analytic elements in a certain kind of sets called quasi-connected sets, as the uniform limit of a sequence of rational functions with no pole in such a set. Later, it appeared that such analytic elements may also be defined in a more general class of sets, particularly infraconnected sets. Recall that a subset of IK is said to be infraconnected if for every a ∈ D, the closure of the image of the function I a defined in D as I a (x) = |x − a|, is an interval. Given a closed bounded subset D of IK, we denote by D the smallest closed disk containing D i.e. if R = diam(D), then D = d(a, R) with a ∈ D. 0
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https://hal.uca.fr/hal-01921872
Contributor : Alain Escassut <>
Submitted on : Wednesday, November 14, 2018 - 10:35:36 AM
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• HAL Id : hal-01921872, version 1

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Alain Escassut, Ta Thi Hoai An. Classical p-adic Nevanlinna theory and Nevalinna Theory out of a hole. Contemporary mathematics, American Mathematical Society, 2018. ⟨hal-01921872⟩

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