Polynomial equations with one catalytic variable, algebraic series and map enumeration

  • Authors:

  • Mireille Bousquet-Mélou;Arnaud Jehanne

  • Affiliations:

  • LaBRI, Université Bordeaux, Talence cedex, France;A2X, Université Bordeaux, Talence cedex, France

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2006

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Abstract

Let F(t, u) = F(u) be a formal power series in t with polynomial coefficients in u. Let F1,...,Fk be k formal power series in t, independent of u. Assume all these series are characterized by a polynomial equation P(F(u), F1,...,Fk,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.