{
  "id": "math/0602478",
  "version": "v2",
  "published": "2006-02-21T21:46:47.000Z",
  "updated": "2006-10-06T22:17:15.000Z",
  "title": "The expected number of zeros of a random system of $p$-adic polynomials",
  "authors": [
    "Steven N. Evans"
  ],
  "comment": "13 pages, no figures, revised to incorporate referees' comments",
  "categories": [
    "math.PR",
    "math.AC"
  ],
  "abstract": "We study the simultaneous zeros of a random family of $d$ polynomials in $d$ variables over the $p$-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the $d$-fold Cartesian product of the $p$-adic integers. Considering models in which the maximum degree that each variable appears is $N$, this expected value is \\[ p^{d \\lfloor \\log_p N \\rfloor} (1 + p^{-1} + p^{-2} + ... + p^{-d})^{-1} \\] for the simplest such model.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-10-06T22:17:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B99",
      "30G15",
      "11S80",
      "30G06"
    ],
    "keywords": [
      "expected number",
      "random system",
      "adic polynomials",
      "fold cartesian product",
      "adic numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2478E"
    }
  }
}