# Rational values of transcendental functions and arithmetic dynamics

### Gareth Boxall

Stellenbosch University, South Africa### Gareth A. Jones

University of Manchester, UK### Harry Schmidt

University Basel, Switzerland

## 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 $\varepsilon$, an upper bound of the form $cD^{3n/4 + \varepsilon n}$ 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 $D\geq 2$ 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 $\varepsilon$.

## Cite this article

Gareth Boxall, Gareth A. Jones, Harry Schmidt, Rational values of transcendental functions and arithmetic dynamics. J. Eur. Math. Soc. 24 (2022), no. 5, pp. 1567–1592

DOI 10.4171/JEMS/1120