Nilpotent polynomials over $\mathbb{Z}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913402254786560 |
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| author | Sengupta, Sayak |
| author_facet | Sengupta, Sayak |
| contents | For a polynomial $u(x)$ in $\mathbb{Z}[x]$ and $r\in\mathbb{Z}$, we consider the orbit of $u$ at $r$ denoted and defined by $\mathcal{O}_u(r):=\{u^{(n)}(r)~|~n\in\mathbb{N}\}$. Here we study polynomials for which $0$ is in the orbit, and we call such polynomials \textit{nilpotent at }$r$ of index $m$ where $m$ is the minimum element of the set $\{n\in\mathbb N~|~u^{(n)}(r)=0\}$. We provide here a complete classification of these polynomials when $|r|\le 4$, with $|r|\le 1$ already covered in the author's previous paper, titled \textit{Locally nilpotent polynomials over $\mathbb Z$}. The central goal of this paper is to study the following questions: (i) relation between the integers $r$ and $m$ when the set of nilpotent polynomials at $r$ of index $m$ is non-empty, (ii) classification of the integer polynomials with nilpotency index $|r|$ for large enough $|r|$, and (iii) bounded integer polynomial sequences $\{r_n\}_{n\ge 0}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01435 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nilpotent polynomials over $\mathbb{Z}$ Sengupta, Sayak Number Theory 11A41, 37P05 (Primary), 11A05, 11A07, 37P25 (Secondary) For a polynomial $u(x)$ in $\mathbb{Z}[x]$ and $r\in\mathbb{Z}$, we consider the orbit of $u$ at $r$ denoted and defined by $\mathcal{O}_u(r):=\{u^{(n)}(r)~|~n\in\mathbb{N}\}$. Here we study polynomials for which $0$ is in the orbit, and we call such polynomials \textit{nilpotent at }$r$ of index $m$ where $m$ is the minimum element of the set $\{n\in\mathbb N~|~u^{(n)}(r)=0\}$. We provide here a complete classification of these polynomials when $|r|\le 4$, with $|r|\le 1$ already covered in the author's previous paper, titled \textit{Locally nilpotent polynomials over $\mathbb Z$}. The central goal of this paper is to study the following questions: (i) relation between the integers $r$ and $m$ when the set of nilpotent polynomials at $r$ of index $m$ is non-empty, (ii) classification of the integer polynomials with nilpotency index $|r|$ for large enough $|r|$, and (iii) bounded integer polynomial sequences $\{r_n\}_{n\ge 0}$. |
| title | Nilpotent polynomials over $\mathbb{Z}$ |
| topic | Number Theory 11A41, 37P05 (Primary), 11A05, 11A07, 37P25 (Secondary) |
| url | https://arxiv.org/abs/2401.01435 |