BOUSQUET-MÉLOU, Mireille; JEHANNE, Arnaud

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BOUSQUET-MÉLOU, Mireille
Laboratoire Bordelais de Recherche en Informatique [LaBRI]

JEHANNE, Arnaud
Théorie des Nombres et Algorithmique Arithmétique [A2X]

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Article de revue

Ce document a été publié dans

Journal of Combinatorial Theory, Series B. 2006, vol. 96, p. 623--672.

Elsevier

Résumé en anglais

Let $F(t,u)\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F_1 , \ldots, F_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$ P(F(u), F_1, \ldots , F_k, t , u)=0. $$ 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.< Réduire

Mots clés en anglais

generating functions

enumeration

kernel method

planar maps

functional equations

quadratic method

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251