Recurrent Inversion Formulas

الأصل · EN

Let F(z)=z-H(z) with o(H(z))≥ 2 be a formal map from ⁿ to ⁿ and G(z) the formal inverse of F(z). In this paper, we fist study the deformation Fₜ(z)=z-tH(z) and its formal inverse map Gₜ(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≥ 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.