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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.20264 |
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| _version_ | 1866911610179682304 |
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| author | Padhy, R. Rout, S. S. |
| author_facet | Padhy, R. Rout, S. S. |
| contents | Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ containing all the archimedean places. It is known from Silverman's result that a forward orbit of a rational map $φ$ contains finitely many $S$-integers in the number field K when $φ^2$ is not a polynomial. Sookdeo stated an analogous conjecture for the backward orbits of a rational map $φ$ using a general $S$-integrality notion based on the Galois conjugates of points. He proved his conjecture for the power map $φ(z) =z^d$ for $d \geq 2$ and consequently for Chebyshev maps (J. Number Theory 131 (2011), 1229-1239). In this paper, we establish uniform bounds on the number of $S$-integral points in the backward orbits of any non-zero $β$ in $K$, relative to a non-preperiodic point $α\in \mathbb{P}^1(\overline{K})$, under the power map $φ(z) =z^d $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20264 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform bounds on $S$-integral points in backward orbits Padhy, R. Rout, S. S. Number Theory Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ containing all the archimedean places. It is known from Silverman's result that a forward orbit of a rational map $φ$ contains finitely many $S$-integers in the number field K when $φ^2$ is not a polynomial. Sookdeo stated an analogous conjecture for the backward orbits of a rational map $φ$ using a general $S$-integrality notion based on the Galois conjugates of points. He proved his conjecture for the power map $φ(z) =z^d$ for $d \geq 2$ and consequently for Chebyshev maps (J. Number Theory 131 (2011), 1229-1239). In this paper, we establish uniform bounds on the number of $S$-integral points in the backward orbits of any non-zero $β$ in $K$, relative to a non-preperiodic point $α\in \mathbb{P}^1(\overline{K})$, under the power map $φ(z) =z^d $. |
| title | Uniform bounds on $S$-integral points in backward orbits |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.20264 |