Explicit Implicit Function Theorem for All Fields

Remark 1. The conditions P (X, f(X)) = 0 and f(0) = 0 imply that P (0, 0) = 0. As P ′ Y (0, 0) is also 0, the sums in both expressions of [X]f are finite. Remark 2. When P (X,Y ) = Xφ(Y ), where φ(X) ∈ K[[X ]] and φ(0) 6= 0, we obtain the Lagrange inversion formula [X]f = [Y ](φ(X) − Y φ(X)φ(X)). If the characteristic of K is 0, we also have the following form [X]f = [Y ]φ(Y ). Remark 3. When P (X,Y ) is a polynomial in X and Y , we obtain a generalisation of FlajoletSoria coefficient extraction formula [2]. Definition 1. Let P (X,Y ) ∈ K[[X,Y ]],