$S$-Integral Points in Orbits on $\mathbb{P}^1$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910004618985472 |
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| author | Yap, Jit Wu |
| author_facet | Yap, Jit Wu |
| contents | Let $K$ be a number field and $S$ a finite set of places of $K$ that contains all of the archimedean places. Let $φ: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $d \geq 2$ defined over $K$. Given $α\in \mathbb{P}^1(K)$ non-preperiodic and $β\in \mathbb{P}^1(K)$ non-exceptional, we prove an upper bound of the form $O(|S|^{1+ε})$ on the number of points in the forward orbit of $α$ that are $S$-integral relative to $β$, extending results of Hsia--Silverman [HS11]. We also prove uniform bounds when $φ$ is a polynomial, extending resaults of Krieger et al [KLS+15]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05094 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $S$-Integral Points in Orbits on $\mathbb{P}^1$ Yap, Jit Wu Number Theory Dynamical Systems Let $K$ be a number field and $S$ a finite set of places of $K$ that contains all of the archimedean places. Let $φ: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $d \geq 2$ defined over $K$. Given $α\in \mathbb{P}^1(K)$ non-preperiodic and $β\in \mathbb{P}^1(K)$ non-exceptional, we prove an upper bound of the form $O(|S|^{1+ε})$ on the number of points in the forward orbit of $α$ that are $S$-integral relative to $β$, extending results of Hsia--Silverman [HS11]. We also prove uniform bounds when $φ$ is a polynomial, extending resaults of Krieger et al [KLS+15]. |
| title | $S$-Integral Points in Orbits on $\mathbb{P}^1$ |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2312.05094 |