{
  "id": "1301.2045",
  "version": "v3",
  "published": "2013-01-10T07:58:37.000Z",
  "updated": "2014-01-17T19:07:10.000Z",
  "title": "Integral-valued polynomials over the set of algebraic integers of bounded degree",
  "authors": [
    "Giulio Peruginelli"
  ],
  "comment": "keywords: Integer-valued polynomial, Algebraic integers with bounded degree, Pr\\\"ufer domain, Polynomially dense subset, Integral closure. To appear in Journal of Number Theory",
  "doi": "10.1016/j.jnt.2013.11.007",
  "categories": [
    "math.NT",
    "math.RA"
  ],
  "abstract": "Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\\in K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)\\subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $f\\in\\mathbb{Q}[X]$: if $f(\\alpha)$ is integral over $\\mathbb{Z}$ for every algebraic integer $\\alpha$ of degree $n$, then $f(\\beta)$ is integral over $\\mathbb{Z}$ for every algebraic integer $\\beta$ of degree smaller than $n$. This second result is established by proving that the integral closure of the ring of polynomials in $\\mathbb{Q}[X]$ which are integer-valued over the set of matrices $M_n(\\mathbb{Z})$ is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to $n$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-01-17T19:07:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13B25",
      "13F20"
    ],
    "keywords": [
      "algebraic integer",
      "integral-valued polynomials",
      "bounded degree",
      "number field",
      "similar property holds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.2045P"
    }
  }
}