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

Let F(t,u)≡F(u)F(t,u)≡F(u) be a formal power series in t with polynomial coefficients in u  . Let F1,…,FkF1,…,Fk be k formal power series in t, independent of u. Assume all these series are characterized by a polynomial equationP(F(u),F1,…,Fk,t,u)=0.P(F(u),F1,…,Fk,t,u)=0. We prove that, under a mild hypothesis on the form of this equation, these k+1k+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)F(u). Applications include the solution of numerous map enumeration problems, among which the hard-particle model on general planar maps.