Nilpotent polynomials over $\mathbb{Z}$

Abstract

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 $\{r_n\}_{n\ge 0}$.