Recurrent Inversion Formulas

Let F (z) = z − H(z) with o(H(z)) ≥ 2 be a formal map from Cn to Cn and G(z) the formal inverse of F (z). In this paper, we give two recurrent formulas for the formal inverse G(z). The first formula not only provides an efficient method for the calculation of G(z), but also reduces the inversion problem to a Cauchy problem of a partial differential equation. The second one is differential free but only works when H(z) is homogeneous of degree d ≥ 2. When d = 3, the second formula coincides with Drużkowski’s formula in [D2]. Some consequences to the well known Jacobian conjecture are also discussed.