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

Let F ( t , u ) ≡ 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.