{
  "id": "1005.4580",
  "version": "v4",
  "published": "2010-05-25T14:07:33.000Z",
  "updated": "2011-03-02T18:33:48.000Z",
  "title": "The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial",
  "authors": [
    "Stavros Garoufalidis"
  ],
  "comment": "16 pages, 4 figures",
  "categories": [
    "math.CO",
    "hep-th",
    "math.GT"
  ],
  "abstract": "A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding holonomic $D$-modules to the case of $q$-holonomic $D$-modules) combined with the Lech-Mahler-Skolem theorem from number theory. En route, we use the Newton polygon of a linear $q$-difference equation, and introduce the notion of regular-singular $q$-difference equation and a WKB basis of solutions of a linear $q$-difference equation at $q=0$. We then use the Lech-Mahler-Skolem theorem to study the vanishing of their leading term. Unlike the case of $q=1$, there are no analytic problems regarding convergence of the WKB solutions.Our proofs are constructive, and they are illustrated by an explicit example.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-03-02T18:33:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "holonomic sequence",
      "quadratic quasi-polynomial",
      "difference equation",
      "lech-mahler-skolem theorem",
      "differential galois theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 891066,
      "adsabs": "2010arXiv1005.4580G"
    }
  }
}