The number of rational iterated preimages of the origin under unicritical polynomial maps
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912651931549696 |
|---|---|
| author | Sano, Kaoru |
| author_facet | Sano, Kaoru |
| contents | We study rational iterated preimages of the origin under unicritical maps $f_{d,c}(x)=x^d+c$. Earlier works of Faber--Hutz--Stoll and Hutz--Hyde--Krause established finiteness and conditional bounds in the quadratic case. Building on this, we prove that for $d=2$ and $c \in \mathbb Q\setminus\{0,-1\}$ there are no rational fourth preimages of the origin, and for all $d \geq 3$ there are no rational second preimages outside trivial cases. The proof relies on geometric analysis of preimage curves, the elliptic Chabauty method, and Diophantine reduction. As a result, we determine the number of rational iterated preimages of $0$ under $f_{d,c}$ for all $d\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The number of rational iterated preimages of the origin under unicritical polynomial maps Sano, Kaoru Number Theory Algebraic Geometry Dynamical Systems 14G05, 14Q05, 37P55 We study rational iterated preimages of the origin under unicritical maps $f_{d,c}(x)=x^d+c$. Earlier works of Faber--Hutz--Stoll and Hutz--Hyde--Krause established finiteness and conditional bounds in the quadratic case. Building on this, we prove that for $d=2$ and $c \in \mathbb Q\setminus\{0,-1\}$ there are no rational fourth preimages of the origin, and for all $d \geq 3$ there are no rational second preimages outside trivial cases. The proof relies on geometric analysis of preimage curves, the elliptic Chabauty method, and Diophantine reduction. As a result, we determine the number of rational iterated preimages of $0$ under $f_{d,c}$ for all $d\geq 2$. |
| title | The number of rational iterated preimages of the origin under unicritical polynomial maps |
| topic | Number Theory Algebraic Geometry Dynamical Systems 14G05, 14Q05, 37P55 |
| url | https://arxiv.org/abs/2510.14397 |