On real solutions of systems of equations

Abstract

Systems of equations f₁ = ··· = f_n−1 = 0 in ℝⁿ = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f₁,..., 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.