## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Alain Escassut

### Growth of an entire function and applications

Abstract

Let $\mathbb{K}$ be an algebraically closed $p$-adic complete field of characteristic zero. We define the order of growth $\rho(f)$ and the type of growth $\sigma(f)$ of an entire function $f(x)= \sum_{n=0}^\infty a_nx^n$ on $\mathbb{K}$ as done on $\mathbb{C}$ and show that $\rho(f)$ and $\sigma(f)$ satisfy the same relations as in complex analysis, with regards to the coefficients $a_n$. But here another expression $\psi(f)$ 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 $\psi(f)=\rho(f)\sigma(f)$. Moreover we show that given a meromorphic function $f={g\over h}$, the number of branched values of $f$ depends on the comparative growth of $g$ and $h$.