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 $\{ r_n\}_{ n\ge 0}$.
Categories: math.NT