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

Gunther, Joseph; Hindes, Wade

doi:10.1090/proc/13653

Integral points of bounded degree on the projective line and in dynamical orbits
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by Joseph Gunther and Wade Hindes PDF

Proc. Amer. Math. Soc. 145 (2017), 5087-5096 Request permission

Abstract:

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 $\varphi (z)\in \overline {\mathbb {Q}}(z)$ be a rational function of degree at least two whose second iterate $\varphi ^2(z)$ 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.

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