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Exponential Formulas for the Jacobians and Jacobian Matrices of Analytic Maps | | Wenhua Zhao
; | | Date: |
24 Sep 2002 | | Journal: | Published in {it J. Pure and Applied Algebra}, extbf{166} (2002) 321-336 | | Subject: | Complex Variables MSC-class: 32H02, 32A05, 14R15 | math.CV | | 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 $A=sum_{i=1}^n a_i(z)frac {p}{z_i}$ 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, ${cal J}(F)=exp (A+ riangledown A)cdot 1$, where $ riangledown A(z)= sum_{i=1}^n frac {a_i}{z_i}(z)$, and $J(F)=exp(A+R_{Ja})cdot I_{n imes n}$, where $I_{n imes n}$ 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 $ riangledown A=0$. Some consequences and applications of the exponential formulas as well as their relations with the well known Jacobian Conjecture are also discussed. | | Source: | arXiv, math.CV/0209312 | | Services: | Forum | Review | PDF | Favorites | |
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