{
  "id": "1705.03709",
  "version": "v1",
  "published": "2017-05-10T11:51:28.000Z",
  "updated": "2017-05-10T11:51:28.000Z",
  "title": "Irreducibility of Random Polynomials",
  "authors": [
    "Christian Borst",
    "Evan Boyd",
    "Claire Brekken",
    "Samantha Solberg",
    "Melanie Matchett Wood",
    "Philip Matchett Wood"
  ],
  "comment": "14 pages, 9 figures",
  "categories": [
    "math.PR",
    "math.NT"
  ],
  "abstract": "We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data supports conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert's Irreducibility Theorem and work of van der Waerden. The data indicates that the probability that a random polynomial is reducible divided by the probability that there is a linear factor appears to approach a constant and, in the large-degree limit, this constant appears to approach one. In cases where the model makes it impossible for the random polynomial to have a linear factor, the probability of reducibility appears to be close to the probability of having a non-linear, low-degree factor. We also study characteristic polynomials of random matrices with +1 and -1 entries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-10T11:51:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11C08",
      "60C05"
    ],
    "keywords": [
      "random polynomial",
      "probability",
      "data supports conjectures",
      "van der waerden",
      "linear factor appears"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}