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Article Dans Une Revue Houston Journal of Mathematics Année : 2014

Growth of p-adic entire functions and applications

Résumé

Let IK be an algebraically closed p-adic complete field of characteristic zero. We define the order of growth ρ(f) and the type of growth σ(f) of an entire function f (x) = ∞ n=0 anx n on IK as done on l C and show that ρ(f) and σ(f) satisfy the same relations as in complex analysis, with regards to the coefficients an. But here another expression ψ(f) that we call cotype of f , depending on the number of zeros inside disks is very important and we show under certain wide hypothesis, that ψ(f) = ρ(f)σ(f), a formula that has no equivalent in complex analysis and suggests that it might hold in the general case. We check that ρ(f) = ρ(f), σ(f) = σ(f) and present an asymptotic relation linking the numbers of zeros inside disks for two functions of same order. That applies to a function and its derivative. We show that the derivative of a transcendental entire function f has infinitely many zeros that are not zeros of f and particularly we show that f cannot divide f when the p-adic absolute value of the number of zeros of f inside disks satisfies certain inequality and particularly when f is of finite order.
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Dates et versions

hal-01920321 , version 1 (13-11-2018)

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  • HAL Id : hal-01920321 , version 1

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Kamal Boussaf, Abdelbaki Boutabaa, Alain Escassut. Growth of p-adic entire functions and applications. Houston Journal of Mathematics, 2014. ⟨hal-01920321⟩
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