Locally nilpotent polynomials over $\mathbb{Z}$

Sayak Sengupta

Locally nilpotent polynomials over $\mathbb{Z}$

Number Theory 2023-09-20 v1 Dynamical Systems

Authors: Sayak Sengupta

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.

Keywords

zeros of polynomials polynomial theory polynomials over finite fields

Cite

@article{arxiv.2309.10303,
  title  = {Locally nilpotent polynomials over $\mathbb{Z}$},
  author = {Sayak Sengupta},
  journal= {arXiv preprint arXiv:2309.10303},
  year   = {2023}
}

Comments

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

Locally nilpotent polynomials over $\mathbb{Z}$

Abstract

Keywords

Cite

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