  We count algebraic points of bounded height and degree on the graphs of
certain functions analytic on the unit disk, obtaining a bound which is
polynomial in the degree and in the logarithm of the multiplicative height. We
combine this work with p-adic methods to obtain a lower bound of the form
$cD^{n/4 - \varepsilon}$ on the degree of the splitting field of $P^{\circ
n}(z)=P^{\circ n}(\alpha)$, where $P$ is a polynomial of degree $D\geq 2$ over
a number field, $P^{\circ n}$ is its $n$-th iterate and $c$ depends effectively
on $P, \alpha$ and $\varepsilon$. Our $c$ is positive for each algebraic
$\alpha$ for which the set $\{P^{\circ n}(\alpha):n\in\mathbb{N}\}$ is
infinite.
