{
  "id": "2309.10303",
  "version": "v1",
  "published": "2023-09-19T04:10:10.000Z",
  "updated": "2023-09-19T04:10:10.000Z",
  "title": "Locally nilpotent polynomials over $\\mathbb{Z}$",
  "authors": [
    "Sayak Sengupta"
  ],
  "comment": "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",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-19T04:10:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A41",
      "37P05",
      "11A05",
      "11A07",
      "37P25"
    ],
    "keywords": [
      "locally nilpotent polynomials",
      "complete classification",
      "special values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}