Locally nilpotent polynomials over $\mathbb{Z}$

TL;DR

This paper classifies integer polynomials based on their iterative behavior at a point, focusing on those that never reach zero but do so modulo every prime, revealing new insights into polynomial orbits over integers.

Contribution

It provides a classification of polynomials over or which zero is not in the orbit but appears modulo all primes, advancing understanding of polynomial dynamics over integers.

Findings

Classified polynomials with zero not in orbit but in all prime moduli

Results for specific points where zero is in the orbit

Insights into polynomial iteration behavior over ields

Abstract

For a polynomial $u(x)$ in $\mathbb{Z}[x]$ and $r\in\mathbb{Z}$, we consider the orbit of $u(x)$ at $r$, $\mathcal{O}_u(r):=\{u(r),u(u(r)),\ldots\}$. 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 classify the polynomials for which (ii) holds. We also present some results for some special $r'$s for which (i) can be answered.