{
  "id": "2211.06760",
  "version": "v1",
  "published": "2022-11-12T22:46:14.000Z",
  "updated": "2022-11-12T22:46:14.000Z",
  "title": "Locally nilpotent polynomials over $\\mathbb{Z}$",
  "authors": [
    "Sayak Sengupta"
  ],
  "comment": "23 pages, 0 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a polynomial $u(x)$ in $\\mathbb{Z}[x]$ and $r\\in\\mathbb{Z}$, we can construct a dynamical sequence $u(r),u^{(2)}(r),\\ldots$. Let $P(u^{(n)}(r)):=\\{p \\mathrm{\\ prime\\ number\\ }|~p \\mathrm{\\ divides\\ }u^{(n)}(r) \\mathrm{\\ for\\ some\\ } n\\}$ and $\\mathcal{P}:=\\{p~|~p \\mathrm{\\ is\\ a\\ prime\\ number}\\}$. For which polynomials $u(x)$ and $r\\in \\mathbb{Z}$ can we say that $P(u^{(n)}(r))=\\mathcal{P}$? If $0$ appears in this sequence, i.e., for some $m\\in\\mathbb{Z}$ $u^{(m)}(r)=0$, then we obviously have the equality $P(u^{(n)}(r))=\\mathcal{P}$. In this paper we study these polynomials. Although the complete classification of such $u(x)'$s for arbitrary $r$ is open, several partial results are proven including the complete classification of $u$ when $r$ is $\\pm 1$ and complete classification of linear polynomials $u(x)$ for arbitrary $r's$ with the extra condition that for no $m\\in\\mathbb{N}$, we get $u^{(m)}(r)=0.$ For the special case when $r$ is a prime we also classify almost all, i.e., all but finitely many linear polynomials $u(x)$ for which we have the equality $P(u^{(n)}(r))=\\mathcal{P}$ and we also give a necessary condition for the remaining finitely many polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-12T22:46:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A41",
      "37P05",
      "11A05",
      "11A07",
      "37P25"
    ],
    "keywords": [
      "locally nilpotent polynomials",
      "complete classification",
      "linear polynomials",
      "necessary condition",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}