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arXiv · 2312.05094

$S$-Integral Points in Orbits on $\mathbb{P}^1$

Jit Wu Yap
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Abstract

Let $K$ be a number field and $S$ a finite set of places of $K$ that contains all of the archimedean places. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $d \geq 2$ defined over $K$. Given $\alpha \in \mathbb{P}^1(K)$ non-preperiodic and $\beta \in \mathbb{P}^1(K)$ non-exceptional, we prove an upper bound of the form $O(|S|^{1+\epsilon})$ on the number of points in the forward orbit of $\alpha$ that are $S$-integral relative to $\beta$, extending results of Hsia--Silverman [HS11]. We also prove uniform bounds when $\varphi$ is a polynomial, extending resaults of Krieger et al [KLS+15].

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Jit Wu Yap. 2023-12-08. $S$-Integral Points in Orbits on $\mathbb{P}^1$. https://arxiv.org/abs/2312.05094

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