Recurrent Inversion Formulas

Wenhua Zhao

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Recurrent Inversion Formulas

Wenhua Zhao

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Abstract

Let $F(z)=z-H(z)$ with $o(H(z))\geq 2$ be a formal map from $\bC^n$ to $\bC^n$ 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 $d\geq 2$. 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.

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Wenhua Zhao. 2004-02-19. Recurrent Inversion Formulas. https://arxiv.org/abs/math/0305162

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