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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.16179 |
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| _version_ | 1866915067727970304 |
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| author | Ghioca, Dragos |
| author_facet | Ghioca, Dragos |
| contents | In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_λ(x):=x^d+λ$ (parameterized by $λ\in\mathbb{C}$), given two starting points $a$ and $b$ in $\mathbb{C}$, if there exist infinitely many $λ\in\mathbb{C}$ such that both $a$ and $b$ are preperiodic under the action of $f_λ$, then $a^d=b^d$. In this paper we study the same question, this time working in a field of characteristic $p>0$. The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points $a$ and $b$ live in $\Fpbar$; (ii) $d$ is a power of $p$; and (iii) not both $a$ and $b$ live in $\Fpbar$, while $d$ is not a power of $p$. Only in case~(iii), one derives the same conclusion as in characteristic $0$, i.e., that $a^d=b^d$. In case~(i), one has that for each $λ\in\Fpbar$, both $a$ and $b$ are preperiodic under the action of $f_λ$, while in case~(ii), one obtains that \emph{also} whenever $a-b\in\Fpbar$, then for each parameter $λ$, we have that $a$ is preperiodic under the action of $f_λ$ if and only if $b$ is preperiodic under the action of $f_λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simultaneously preperiodic points for a family of polynomials in positive characteristic Ghioca, Dragos Number Theory In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_λ(x):=x^d+λ$ (parameterized by $λ\in\mathbb{C}$), given two starting points $a$ and $b$ in $\mathbb{C}$, if there exist infinitely many $λ\in\mathbb{C}$ such that both $a$ and $b$ are preperiodic under the action of $f_λ$, then $a^d=b^d$. In this paper we study the same question, this time working in a field of characteristic $p>0$. The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points $a$ and $b$ live in $\Fpbar$; (ii) $d$ is a power of $p$; and (iii) not both $a$ and $b$ live in $\Fpbar$, while $d$ is not a power of $p$. Only in case~(iii), one derives the same conclusion as in characteristic $0$, i.e., that $a^d=b^d$. In case~(i), one has that for each $λ\in\Fpbar$, both $a$ and $b$ are preperiodic under the action of $f_λ$, while in case~(ii), one obtains that \emph{also} whenever $a-b\in\Fpbar$, then for each parameter $λ$, we have that $a$ is preperiodic under the action of $f_λ$ if and only if $b$ is preperiodic under the action of $f_λ$. |
| title | Simultaneously preperiodic points for a family of polynomials in positive characteristic |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.16179 |