Nilpotent polynomials over $\mathbb{Z}$

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 ${\mathcal{O}}_u(r):=\{u^{(n)}(r)\mathrm{\mid }n\in \mathbb{N}\}$. Here we study polynomials for which $0$ is in the orbit, and we call such polynomials $\mathit{\text{nilpotent}}\mathit{\text{at}}$$r$ of index $m$ where $m$ is the minimum element of the set $\{n\in \mathbb{N}\mathrm{\mid }{u}^{(n)}(r)=0\}$. We provide here a complete classification of these polynomials when $\mathrm{\mid }r\mathrm{\mid }\le 4$, with $\mathrm{\mid }r\mathrm{\mid }\le 1$ already covered in the author’s previous paper, titled $\mathit{\text{Locally}}\mathit{\text{nilpotent}}\mathit{\text{polynomials}}\mathit{\text{over}}\mathbb{Z}$. 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 $\mathrm{\mid }r\mathrm{\mid }$ for large enough $\mathrm{\mid }r\mathrm{\mid }$, and (iii) bounded integer polynomial sequences $\{r_n{\}}_{n\ge 0}$.