{
  "id": "1510.08807",
  "version": "v1",
  "published": "2015-10-29T18:19:28.000Z",
  "updated": "2015-10-29T18:19:28.000Z",
  "title": "Canonical heights and preperiodic points for subhomogeneous families of polynomials",
  "authors": [
    "Patrick Ingram"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A family $f_t(z)$ of polynomials over a number field $K$ will be called \\emph{subhomogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\\geq 2$. For example, the family $z^d+t$ is subhomogeneous. We prove a lower bound on the canonical height, of the form \\[\\hat{h}_{f_t}(z)\\geq \\epsilon \\max\\{h_{\\mathsf{M}_d}(f_t), \\log|\\operatorname{Norm}\\mathfrak{R}_{f_t}|\\},\\] for values $z\\in K$ which are not preperiodic for $f_t$. Here $\\epsilon$ depends only on the number of places at which $f_t$ has bad reduction. For suitably generic morphisms $\\varphi:\\mathbb{P}^1\\to \\mathbb{P}^1$, we also prove an absolute bound of this form for $t$ in the image of $\\varphi$ over $K$ (assuming the $abc$ Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-29T18:19:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P30"
    ],
    "keywords": [
      "preperiodic points",
      "canonical height",
      "subhomogeneous families",
      "polynomials",
      "bad reduction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151008807I"
    }
  }
}