Integral points of bounded degree on the projective line and in dynamical orbits

Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $φ(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $φ^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.

No reviews yet.

No comments yet.