{
  "id": "2103.06163",
  "version": "v1",
  "published": "2021-03-10T16:22:05.000Z",
  "updated": "2021-03-10T16:22:05.000Z",
  "title": "On Certain Probabilistic Properties of Polynomials over the Ring of $p$-adic Integers",
  "authors": [
    "Antonio Lei",
    "Antoine Poulin"
  ],
  "journal": "American Mathematical Monthly, 127 (6), 2020, 519-529",
  "doi": "10.1080/00029890.2020.1736468",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo powers of $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-10T16:22:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic integers",
      "probabilistic properties",
      "monic polynomial",
      "strongly coprime",
      "integers modulo powers"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Taylor-Francis"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}