Recurrent Inversion Formulas

Let $F(z)=z-H(z)$ with $o(H(z))\ge 2$ be a formal map from ${\text{bC}}^n$ to ${\text{bC}}^n$ and $G(z)$ the formal inverse of $F(z)$. In this paper, we fist study the deformation $F_t(z)=z-tH(z)$ and its formal inverse map $G_t(z)$. We then derive two recurrent formulas for the formal inverse $G(z)$. The first formula in certain situations provides a more efficient method for the calculation of $G(z)$ than other well known inversion formulas. The second one is differential free but only works when $H(z)$ is homogeneous of degree $d\ge 2$. Finally, we reveal a close relationship of the inversion problem with a Cauchy problem of a PDE. When the Jacobian matrix $JF(z)$ is symmetric, the PDE coincides with the $n$-dimensional inviscid Burgers’ equation in Diffusion theory.