Alain Escassut Growth of an entire function and applicationsAbstract
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 $h$.
