{
  "id": "1906.04514",
  "version": "v1",
  "published": "2019-06-11T12:10:45.000Z",
  "updated": "2019-06-11T12:10:45.000Z",
  "title": "Simultaneously preperiodic integers for quadratic polynomials",
  "authors": [
    "Valentin Huguin"
  ],
  "comment": "13 pages, 6 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this article, we study the set of parameters $c \\in \\mathbb{C}$ for which two given complex numbers $a$ and $b$ are simultaneously preperiodic for the quadratic polynomial $f_{c}(z) = z^{2} +c$. Combining complex-analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if $a^{2} = b^{2}$. Recently, Buff answered a question of theirs, proving that the set of parameters $c \\in \\mathbb{C}$ for which both $0$ and $1$ are preperiodic for $f_{c}$ is equal to $\\lbrace -2, -1, 0 \\rbrace$. Following his approach, we complete the description of these sets when $a$ and $b$ are two given integers with $\\lvert a \\rvert \\neq \\lvert b \\rvert$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-11T12:10:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "37F45",
      "37P35"
    ],
    "keywords": [
      "simultaneously preperiodic integers",
      "quadratic polynomial",
      "parameters",
      "complex numbers",
      "arithmetic arguments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}