Locally nilpotent polynomials over $\mathbb{Z}$

Sayak Sengupta

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arXiv · 2211.06760

Locally nilpotent polynomials over $\mathbb{Z}$

Sayak Sengupta

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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.

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Sayak Sengupta. 2022-11-12. Locally nilpotent polynomials over $\mathbb{Z}$. https://arxiv.org/abs/2211.06760

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