递推反演公式
Recurrent Inversion Formulas
摘要：
设具有$o(H(z))\geq 2$的$F(z)=z-H(z)$是从$\bC^n$到$\bC^n$的形式映射，而$G(z)$是$F(z)$的形式逆。本文首先研究了变形$F_t(z)=z-tH(z)$及其形式逆映射$G_t(z)$。然后，我们推导出两个递归公式的形式逆$G(z)$。在某些情况下，第一个公式提供了一种比其它已知反演公式更有效的计算$G(z)$的方法。第二个是微分自由的，但只有当$H(z)$是齐次$d\geq 2$。最后，我们揭示了反演问题与偏微分方程柯西问题的密切关系。当雅可比矩阵$JF(z)$对称时，PDE与扩散理论中的$n$维无粘Burgers方程一致.
英文摘要：
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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