反演问题、Legendre变换和无粘Burgers方程
Inversion Problem, Legendre Transform and Inviscid Burgers' Equations
摘要：
令序为$o(H(z))\geq 1$的$F(z)=z-H(z)$是从$\bC^n$到$\bC^n$的形式映射，$G(z)$是$F(z)$的形式逆映射.首先研究了$F(z)$的变形$F_t(z)=z-tH(z)$及其形式逆$G_t(z)=z+tN_t(z)$。(Note当$o(H(z))\geq 2$时，$G_{t=1}(z)=G(z)$。）我们表明，$N_t(z)$是唯一的幂级数解的柯西问题的偏微分方程，从中我们推导出一个经常性公式$G_t(z)$。其次，在M. de Bondt，A. van den Essen \cite{BE1}和G.在Jacobian猜想的基础上，我们考虑满足梯度条件的形式映射$F(z)=z-H(z)$，即对于某些$o(P(z))\geq 2$阶的$P(z)\in \bC[[z]]$，有$H(z)=\nabla P(z)$.证明了在梯度条件下，$N_t(z)=\nabla Q_t(z)$对某些$Q_t(z)\in \bC[[z, t]]$和$N_t(z)$所满足的PDE成为$n$维无粘Burgers方程，并由此导出了$Q_t(z)$的递推公式.进一步阐明了反演问题、Legendre变换和无粘Burgers方程之间的密切关系。特别是雅可比猜想减少到一个无粘Burgers方程的问题。最后，在梯度条件下，推导出$Q_t(z)$的二叉树展开反演公式。本文导出的递推反演公式和二元根树展开反演公式也可用作求解无粘Burgers方程的某些Cauchy问题和$o(f(z))\geq 2$的幂级数$f(z)$的Legendre变换的计算算法。
英文摘要：
Let $F(z)=z-H(z)$ with order $o(H(z))\geq 1$ be a formal map from $\bC^n$ to $\bC^n$ and $G(z)$ the formal inverse map of $F(z)$. We first study the deformation $F_t(z)=z-tH(z)$ of $F(z)$ and its formal inverse $G_t(z)=z+tN_t(z)$. (Note that $G_{t=1}(z)=G(z)$ when $o(H(z))\geq 2$.) We show that $N_t(z)$ is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for $G_t(z)$. Secondly, motivated by the gradient reduction obtained by M. de Bondt, A. van den Essen \cite{BE1} and G. Meng \cite{M} for the Jacobian conjecture, we consider the formal maps $F(z)=z-H(z)$ satisfying the gradient condition, i.e. $H(z)=\nabla P(z)$ for some $P(z)\in \bC[[z]]$ of order $o(P(z))\geq 2$. We show that, under the gradient condition, $N_t(z)=\nabla Q_t(z)$ for some $Q_t(z)\in \bC[[z, t]]$ and the PDE satisfied by $N_t(z)$ becomes the $n$-dimensional inviscid Burgers' equation, from which a recurrent formula for $Q_t(z)$ also follows. Furthermore, we clarify some close relationships among the inversion problem, Legendre transform and the inviscid Burgers' equations. In particular the Jacobian conjecture is reduced to a problem on the inviscid Burgers' equations. Finally, under the gradient condition, we derive a binary rooted tree expansion inversion formula for $Q_t(z)$. The recurrent inversion formula and the binary rooted tree expansion inversion formula derived in this paper can also be used as computational algorithms for solutions of certain Cauchy problems of the inviscid Burgers' equations and Legendre transforms of the power series $f(z)$ of $o(f(z))\geq 2$.
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