{
  "id": "1402.4628",
  "version": "v1",
  "published": "2014-02-19T11:50:57.000Z",
  "updated": "2014-02-19T11:50:57.000Z",
  "title": "On the number of real roots of random polynomials",
  "authors": [
    "Hoi Nguyen",
    "Oanh Nguyen",
    "Van Vu"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the expectation of the number of real roots is $\\frac{2}{\\pi} \\log n + o(\\log n)$. In this paper, we determine the true nature of the error term by showing that the expectation equals $\\frac{2}{\\pi}\\log n + O(1)$. Prior to this paper, such estimate has been known only in the gaussian case, thanks to works of Edelman and Kostlan.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-19T11:50:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real roots",
      "random polynomials",
      "generalizing earlier fundamental works",
      "error term",
      "long time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.4628N"
    }
  }
}