{
  "id": "2401.01435",
  "version": "v1",
  "published": "2024-01-02T21:10:49.000Z",
  "updated": "2024-01-02T21:10:49.000Z",
  "title": "Nilpotent polynomials over $\\mathbb{Z}$",
  "authors": [
    "Sayak Sengupta"
  ],
  "comment": "19 pages in total. This is a continuation of the research done by the author in the paper \"Locally nilpotent polynomials over $\\mathbb{Z}$\". Comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a polynomial $u(x)$ in $\\mathbb{Z}[x]$ and $r\\in\\mathbb{Z}$, we consider the orbit of $u$ at $r$ denoted and defined by $\\mathcal{O}_u(r):=\\{u^{(n)}(r)~|~n\\in\\mathbb{N}\\}$. Here we study polynomials for which $0$ is in the orbit for a given $r$. We provide a complete classification of these polynomials when $|r|\\le 4$, with $|r|\\le 1$ already done in \\cite{SS23}. The central goal of this paper is to study the following questions: (i) relationship between the integers $r$ and $m$, for a polynomial $u$ in $N_{r,m}$; (ii) classification of the polynomials with nilpotency index $|r|$ for large enough $|r|$; and (iii) integer sequences with a generating polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-02T21:10:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A41",
      "37P05",
      "11A05",
      "11A07",
      "37P25"
    ],
    "keywords": [
      "nilpotent polynomials",
      "study polynomials",
      "complete classification",
      "integer sequences",
      "central goal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}