{
  "id": "1710.04332",
  "version": "v1",
  "published": "2017-10-12T00:22:21.000Z",
  "updated": "2017-10-12T00:22:21.000Z",
  "title": "Riccati equations and polynomial dynamics over function fields",
  "authors": [
    "Wade Hindes",
    "Rafe Jones"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a function field $K$ and $\\phi \\in K[x]$, we study two finiteness questions related to iteration of $\\phi$: whether all but finitely many terms of an orbit of $\\phi$ must possess a primitive prime divisor, and whether the Galois groups of iterates of $\\phi$ must have finite index in their natural overgroup $\\text{Aut}(T_d)$, where $T_d$ is the infinite tree of iterated preimages of $0$ under $\\phi$. We focus particularly on the case where $K$ has characteristic $p$, where far less is known. We resolve the first question in the affirmative under relatively weak hypotheses; interestingly, the main step in our proof is to rule out \"Riccati differential equations\" in backwards orbits. We then apply our result on primitive prime divisors and adapt a method of Looper to produce a family of polynomials for which the second question has an affirmative answer; these are the first non-isotrivial examples of such polynomials. We also prove that almost all quadratic polynomials over $\\mathbb{Q}(t)$ have iterates whose Galois group is all of $\\text{Aut}(T_d)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-12T00:22:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "function field",
      "riccati equations",
      "polynomial dynamics",
      "primitive prime divisor",
      "galois group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}