Exponential Formulas for the Jacobians and Jacobian Matrices of Analytic Maps

Abstract

Let $F=(F_1, F_2, ... F_n)$ be an $n$-tuple of formal power series in $n$ variables of the form $F(z)=z+ O(|z|^2)$. It is known that there exists a unique formal differential operator $A=\sum_{i=1}^n a_i(z)\frac {\p}{\p z_i}$ such that $F(z)=exp (A)z$ as formal series. In this article, we show the Jacobian ${\cal J}(F)$ and the Jacobian matrix $J(F)$ of $F$ can also be given by some exponential formulas. Namely, ${\cal J}(F)=\exp (A+\triangledown A)\cdot 1$, where $\triangledown A(z)= \sum_{i=1}^n \frac {\p a_i}{\p z_i}(z)$, and $J(F)=\exp(A+R_{Ja})\cdot I_{n\times n}$, where $I_{n\times n}$ is the identity matrix and $R_{Ja}$ is the multiplication operator by $Ja$ for the right. As an immediate consequence, we get an elementary proof for the known result that ${\cal J}(F)\equiv 1$ if and only if $\triangledown A=0$. Some consequences and applications of the exponential formulas as well as their relations with the well known Jacobian Conjecture are also discussed.