Locally 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 $u(x)$ at $r$, ${\mathcal{O}}_u(r):=\{u(r),u(u(r)),\dots \}$. We ask two questions here: (i) what are the polynomials $u$ for which $0\in {\mathcal{O}}_u(r)$ and (ii) what are the polynomials for which $0\notin {\mathcal{O}}_u(r)$ but, modulo every prime $p$, $0\in {\mathcal{O}}_u(r)$? In this paper we classify the polynomials for which (ii) holds. We also present some results for some special $r^{\prime }$s for which (i) can be answered.

18 pages, 0 figures. Comments are welcome!