Locally nilpotent polynomials over $\mathbb{Z}$

Abstract

For a polynomial $u=u(x)$ in $\mathbb{Z}[x]$ and $r\in \mathbb{Z}$, we consider the orbit of $u$ at $r$ denoted and defined by ${\mathcal{O}}_u(r):=\{u(r),u(u(r)),\dots \}$. 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 give a complete classification of the polynomials for which (ii) holds for a given $r$. We also present some results for some special values of $r$ where (i) can be answered.

17 pages. Few edits throughout the entire paper and one newly added subsection. To appear in INTEGERS. arXiv admin note: substantial text overlap with arXiv:2211.06760