{
  "id": "1508.07624",
  "version": "v1",
  "published": "2015-08-30T19:37:22.000Z",
  "updated": "2015-08-30T19:37:22.000Z",
  "title": "Some finiteness results on monogenic orders in positive characteristic",
  "authors": [
    "Jason P. Bell",
    "Khoa D. Nguyen"
  ],
  "comment": "27 pages, comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let $\\mathcal{O}$ is a finitely generated $\\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix $s$ that is integral over $\\mathcal{O}$, describe all $t$ such that $\\mathcal{O}[s]=\\mathcal{O}[t]$. (B) Fix $s$ and $t$ that are integral over $\\mathcal{O}$, describe all pairs $(m,n)\\in\\mathbb{N}^2$ such that $\\mathcal{O}[s^m]=\\mathcal{O}[t^n]$. In this paper, we solve these problems and provide a uniform bound for a certain \"discriminant form equation\" that is closely related to Problem (A) when $\\mathcal{O}$ has characteristic $p>0$. While our general strategy roughly follows [EG85] and [Ngu15], many new delicate issues arise due to the presence of the Frobenius automorphisms $x\\mapsto x^p$. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-30T19:37:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11R99",
      "11T99"
    ],
    "keywords": [
      "positive characteristic",
      "monogenic orders",
      "finiteness results",
      "discriminant form equation",
      "delicate issues arise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150807624B"
    }
  }
}