On real solutions of systems of equations

Systems of equations f ₁ = ··· = f _n?1 = 0 in ? ⁿ = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f ₁,..., f _n?1 are independent at x ≠ 0. It is shown that, for n ≠ 2 and n ≠ 4, such a system has a smooth solution which passes through x = 0 and has nonzero Maclaurin series.