{
  "id": "2007.14567",
  "version": "v1",
  "published": "2020-07-29T03:12:28.000Z",
  "updated": "2020-07-29T03:12:28.000Z",
  "title": "Irreducibility of random polynomials: general measures",
  "authors": [
    "Lior Bary-Soroker",
    "Dimitris Koukoulopoulos",
    "Gady Kozma"
  ],
  "comment": "57 pages",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "In this paper we prove that if the coefficients of a monic polynomial $f(x)\\in\\mathbb{Z}[x]$ of degree $n$ are chosen independently at random according to measures $\\mu_0,\\mu_1,\\dots,\\mu_{n-1}$ whose support is sufficiently large, then $f(x)$ is irreducible with probability tending to $1$ as $n$ tends to infinity. In particular, we prove that if $f(x)$ is a randomly chosen polynomial of degree $n$ and coefficients in $\\{1,2,\\dots,H\\}$ with $H\\ge35$, then it is irreducible with probability tending to 1 as $n$ tends to infinity. More generally, we prove that if we choose the coefficients of $f(x)$ independently and uniformly at random from a set $\\mathcal{N}\\subset[1,H]$ of $\\ge H^{4/5}(\\log H)^2$ integers with $H$ sufficiently large, then $f(x)$ is irreducible with probability tending to $1$ as $n$ tends to infinity. In addition, in all of these settings, we show that the Galois group of $f(x)$ is either $\\mathcal{A}_n$ or $\\mathcal{S}_n$ with high probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-29T03:12:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R09",
      "12E05",
      "11N25",
      "11T55",
      "05A05",
      "20B30"
    ],
    "keywords": [
      "random polynomials",
      "general measures",
      "probability tending",
      "irreducibility",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}