Polynomial equations with one catalytic variable, algebraic series, and map enumeration
Language
en
Article de revue
This item was published in
Journal of Combinatorial Theory, Series B. 2006, vol. 96, p. 623--672.
Elsevier
English Abstract
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 ...Read more >
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.Read less <
English Keywords
generating functions
enumeration
kernel method
planar maps
functional equations
quadratic method
Origin
Hal imported