{
  "id": "2201.11707",
  "version": "v1",
  "published": "2022-01-27T18:06:48.000Z",
  "updated": "2022-01-27T18:06:48.000Z",
  "title": "Polynomials with many rational preperiodic points",
  "authors": [
    "John R. Doyle",
    "Trevor Hyde"
  ],
  "comment": "Comments welcome!",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\\mathbb{Q}[x]$. We show that for all sufficiently large integers $d$, there exists a polynomial $f_d(x) \\in \\mathbb{Q}[x]$ with $2\\leq \\mathrm{deg}(f_d) \\leq d$ such that $f_d(x)$ has at least $d + \\lfloor \\log_{16}(d)\\rfloor$ rational preperiodic points. Furthermore, we show that for infinitely many $d \\geq 2$, the polynomials $f_d(x)$ and $f_d(x) + 1$ have at least $d^2 + d\\lfloor \\log_{16}(d)\\rfloor - 2d + 1$ common complex preperiodic points. To prove the existence of such polynomials we introduce the technique of dynamical compression.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-27T18:06:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational preperiodic points",
      "polynomial",
      "common complex preperiodic points",
      "sufficiently large integers",
      "exceptional behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}