{
  "id": "1610.00807",
  "version": "v1",
  "published": "2016-10-04T00:41:16.000Z",
  "updated": "2016-10-04T00:41:16.000Z",
  "title": "Periodic points of quadratic polynomials in Galois extensions",
  "authors": [
    "Robin Zhang"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "For quadratic polynomials with coefficients in a field $k$ of characteristic zero, we seek to understand the periodic points under the actions of its iteration and the absolute Galois group $\\mathrm{Gal}(\\bar{k}/k)$. Supported by previous numerical work, we conjecture that these two actions are highly correlated. When $k = \\mathbb{Q}$, we show that this conjecture implies the non-existence of quadratic periodic points of period $5$ and prove that this conjecture is true for almost all rational $c$ using Hilbert's irreducibility theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-04T00:41:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G30",
      "14G05",
      "37P35"
    ],
    "keywords": [
      "quadratic polynomials",
      "galois extensions",
      "hilberts irreducibility theorem",
      "quadratic periodic points",
      "absolute galois group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}