Rational values of transcendental functions and arithmetic dynamics

We count algebraic points of bounded height and degree on the graphs certain functions analytic unit disk, obtaining a bound which is polynomial in logarithm multiplicative height. combine this work with $p$-adic methods to obtain, for each positive $\varepsilon$, an upper form $cD^{3n/4 + \varepsilon n}$ number irreducible factors $P^{\circ n}(X)-P^{\circ n}(\alpha)$ over $K$, where $K$ field, $P$ $D\geq 2$ $n$-th iterate $P$, $\alpha$ point ${P^{\circ n}(\alpha):n\in\mathbb{N}}$ infinite $c$ depends effectively $P, \alpha, \[K:\mathbb{Q}]$ $\varepsilon$.