Locally nilpotent polynomials over $\mathbb{Z}$

TL;DR

This paper classifies integer polynomials based on their iterative behavior at a point, distinguishing those whose orbits either include zero or are zero-free but locally zero modulo all primes.

Contribution

It provides a complete classification of polynomials satisfying a specific local-global orbit property over integers and primes.

Findings

Classified polynomials with zero in their orbit at a point.

Characterized polynomials with zero-free orbits locally modulo all primes.

Presented results for special cases of the starting point r.

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 $\mathcal{O}_u(r):=\{u(r),u(u(r)),\ldots\}$. 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 $0\not\in \mathcal{O}_u(r)$ 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.