{
  "id": "2101.03561",
  "version": "v1",
  "published": "2021-01-10T14:52:15.000Z",
  "updated": "2021-01-10T14:52:15.000Z",
  "title": "The Expected Number of Roots over The Field of p-adic Numbers",
  "authors": [
    "Roy Shmueli"
  ],
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "We study the roots of a random polynomial over the field of $p$-adic numbers. For a random monic polynomial with i.i.d.\\ coefficients in $\\mathbb{Z}_p$, we obtain an estimate for the expected number of roots of this polynomial. In particular, if the coefficients take the values $\\pm1$ with equal probability, the expected number of $p$-adic roots converges to $\\left(p-1\\right)/\\left(p+1\\right)$ as the degree of the polynomial tends to $\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-10T14:52:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S05",
      "65H04"
    ],
    "keywords": [
      "expected number",
      "p-adic numbers",
      "random monic polynomial",
      "adic roots converges",
      "random polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}