Rational values of transcendental functions and arithmetic dynamics

Abstract

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, for each positive $\epsilon$, an upper bound of the form $c{D}^{3n/4+\epsilon n}$ on the number of irreducible factors of $P^{\circ n}(X)-P^{\circ n}(\alpha )$ over $K$, where $K$ is a number field, $P$ is a polynomial of degree $D\ge 2$ over $K$, $P^{\circ n}$ is the $n$-th iterate of $P$, $\alpha$ is a point in $K$ for which $\{P^{\circ n}(\alpha ):n\in \mathbb{N}\}$ is infinite and $c$ depends effectively on $P,\alpha ,[K:\mathbb{Q}]$ and $\epsilon$.