{
  "id": "math/0504018",
  "version": "v1",
  "published": "2005-04-01T14:04:07.000Z",
  "updated": "2005-04-01T14:04:07.000Z",
  "title": "Polynomial equations with one catalytic variable, algebraic series, and map enumeration",
  "authors": [
    "Mireille Bousquet-Mélou",
    "Arnaud Jehanne"
  ],
  "journal": "Journal of Combinatorial Theory Series B 96 (2006) 623--672",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $F(t,u)\\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F\\_1, ..., F\\_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$ P(F(u), F\\_1, ..., F\\_k, t, u)=0. $$ We prove that, under a mild hypothesis on the form of this equation, these $(k+1)$ series are algebraic, and we give a strategy to compute a polynomial equation for each of them. This strategy generalizes the so-called kernel method, and quadratic method, which apply respectively to equations that are linear and quadratic in $F(u)$. Applications include the solution of numerous map enumeration problems, among which the hard-particle model on general planar maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-04-01T14:04:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "polynomial equation",
      "algebraic series",
      "catalytic variable",
      "formal power series",
      "general planar maps"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4018B"
    }
  }
}