On solutions to formal equations
Tejinder S. Neelon · Bulletin of the Belgian Mathematical Society - Simon Stevin · 2000
Let k be a field of characteristic zero equipped with an absolute value |•|.Let φ 1 (x, y) = φ 2 (x, y) = . . .= φ l (x, y) = 0 be a system of formal power series equations in variables x = (x 1, . . .x n ), y = (y 1, . . .y m ) with coefficients in k.The notion of {M k }-summability of formal power series is defined relative to a sequence {M k } ∞ k=0 of positive real numbers.Under certain Jacobian conditions on the φ i 's, it is shown the {M k }-summability of the φ i 's implies {M k }-summability of any of its formal power series solutions y = f (x).In particular, if the φ i 's are convergent, then so are its formal solutions.This result generalizes the author's earlier work on formal solutions of systems of analytic equations.