{
  "id": "1708.00944",
  "version": "v1",
  "published": "2017-08-02T21:58:40.000Z",
  "updated": "2017-08-02T21:58:40.000Z",
  "title": "On Multiplicative Independence of Rational Iterates",
  "authors": [
    "Marley Young"
  ],
  "comment": "14 pages, comments appreciated",
  "categories": [
    "math.NT"
  ],
  "abstract": "Lower bounds are given for the degree of multiplicative combinations of iterates of certain classes of rational functions over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao's method for constructing elements in $\\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$ becomes large. Additionally, for a field $\\mathbb{F}$ of characteristic $0$, an upper bound is given for the number of polynomials $u \\in \\mathbb{F}[X]$ such that $\\{ F_i(X,u(X)) \\}_{i=1}^n$ is multiplicatively dependent for given rational functions $F_1,\\ldots,F_n \\in \\mathbb{F}(X,Y)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-02T21:58:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiplicative independence",
      "rational iterates",
      "rational functions",
      "lower bounds",
      "general field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}