{
  "id": "0706.2166",
  "version": "v1",
  "published": "2007-06-14T17:38:45.000Z",
  "updated": "2007-06-14T17:38:45.000Z",
  "title": "Canonical heights and the arithmetic complexity of morphisms on projective space",
  "authors": [
    "Shu Kawaguchi",
    "Joseph H. Silverman"
  ],
  "comment": "submitted to the Quarterly Journal of Pure and Applied Mathematics",
  "journal": "Pure and Applied Mathematics Quarterly 5 (2009), 1201--1217",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let F and G be morphisms of degree at least 2 from P^N to P^N that are defined over the algebraic closure of Q. We define the arithmetic distance d(F,G) between F and G to be the supremum over all algebraic points P of |h_F(P)-h_G(P)|, where h_F and h_G are the canonical heights associated to the morphisms F and G, respectively. We prove comparison theorems relating d(F,G) to more naive height functions and show that for a fixed G, the set of F satisfying d(F,G) < B is a set of bounded height. In particular, there are only finitely many such F defined over any given number field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-14T17:38:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "14G40",
      "37F10"
    ],
    "keywords": [
      "canonical heights",
      "arithmetic complexity",
      "projective space",
      "algebraic closure",
      "arithmetic distance"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.2166K"
    }
  }
}