Order, Type and Cotype of Growth for p-adic Entire Function

Abstract

Let $\K$ be a complete ultrametric algebraically closed field and let ${\cal A}(\K)$ be the $\K$-algebra of entires functions on $\K$. For an $f\in {\cal A}(\K)$, similarly to complex analysis, one can define the order of growth as $\rho(f)=\dsp{\limsup_{r\to +\infty}{\log(\log(|f|(r))\over \log r}}$. When $\rho(f)\neq 0, +\infty$, one can define the type of growth as $\sigma(f)=\dsp{\limsup_{r\to +\infty}{\log(|f|(r))\over r^{\rho(f)}}}$. But here, we can also define the cotype of growth as $\psi(f)=\dsp{\limsup_{r\to +\infty}{q(f,r)\over r^{\rho(f)}}}$ where $q(f,r)$ is the number of zeros of $f$ in the disk of center $0$ and radius $r$. Then we have $\rho(f)\sigma(f)\leq \psi(f)\leq e\rho(f)\sigma(f)$. Moreover, if $\psi$ or $\sigma$ are veritable limits, then $\rho(f)\sigma(f)= \psi(f)$ and this relation is conjectured in the general case. Many other properties are examined concerning $\rho(f),\ \sigma(f),\ \psi(f)$. Particularly, we prove that if an entire function $f$ has finite order, then $\dsp{f'\over f^2}$ takes every value infinitely many times and applications are shown to branched values of meromorphic functions.

^* This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.