Exponential formulas for the Jacobians and Jacobian matrices of analytic maps

Let F=(F1,F2,…,Fn) be an n-tuple of formal power series in n variables of the form F(z)=z+O(z2). It is known that there exists a unique formal differential operator such that F(z)=exp(A)z as formal series. In this article, we show the Jacobian and the Jacobian matrix J(F) of F can also be given by some exponential formulas. Namely, , where , and J(F)=exp(A+RJa)•In×n, where In×n is the identity matrix and RJa is the multiplication operator by Ja for the right. As an immediate consequence, we get an elementary proof for the known result that if and only if A=0. Some consequences and applications of the exponential formulas as well as their relations with the well-known Jacobian Conjecture are also discussed.