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(|z|). It is known that, there exists a unique formal differential operator A(z) = ∑n i=1 ai(z) ∂ ∂zi such that F (z) = exp(A)z as formal series. In this article, we show the Jacobian J(F ) and the Jacobian matrix J(F ) of F can also be given by some exponential formulas. Namely, J(F ) = exp(A+▽A) ·1, where ▽A(z) = ∑n i=1 ∂ai ∂zi (z), 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 J(F ) ≡ 1 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.