Order, type and cotype of growth for p-adic entire functions

Abstract : 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}, by C(α, r) the circle {x ∈ IK | |x − α| = r}, by A(IK) the IK-algebra of analytic functions in IK (i.e. the set of power series with an infinite radius of convergence) and by M(IK) the field of meromorphic functions in IK (i.e. the field of fractions of A(IK)). Given f ∈ M(IK), we will denote by q(f, r) the number of zeros of f in d(0, r), taking multiplicity into account and by u(f, r) the number of distinct multiple zeros of f in d(0, r). Throughout the paper, log denotes the Neperian logarithm. Here we mean to introduce and study the notion of order of growth and type of growth for functions of order t. We will also introduce a new notion of cotype of growth in relation with the distribution of zeros in disks which plays a major role in processes that are quite different from those in complex analysis. This has an application to the question whether an entire function can be devided by its derivative inside the algebra of entire functions. Let us shortly recall classical results [4], [5], [6]: Theorem A Given f ∈ A(IK) and r > 0, we denote by |f |(r) the number sup{|f (x)| | |x| = r} and then |. |(r) is a multiplicative norm on A(IK). Suppose f (0) = 0 and let a 1 , ..., a m be the varius zeros of f in d(0, r) with |a n | ≤ |a n+1 |, 1 ≤ n ≤ m − 1, each zero a n having a multiplicity order w n. Then log(|f |(r)) = log(|f (0)|) + m n=1 w n (log(r) − log(|a n |)).
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Submitted on : Tuesday, November 13, 2018 - 11:36:52 AM
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Kamal Boussaf, Abdelbaki Boutabaa, Alain Escassut. Order, type and cotype of growth for p-adic entire functions. p-Adic Numbers, Ultrametric Analysis and Applications, MAIK Nauka/Interperiodica, 2018. ⟨hal-01920435⟩



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