Let $f_1,\dots,f_n$ be formal power series in $\mathbb{C}[[x_1,\dots,x_n]]$ whose constant terms are all zero (i.e. $f_1,\dots f_n$ are not units in the ring). Suppose further that the radical of the ideal $(f_1,\dots f_n)$ is the maximal ideal. Does this imply that the Jacobian determinant $$\det \left(\frac{\partial f_i}{\partial x_j}\right)$$ is not identically zero?

**Note**, that in the special case when $f_1,\dots f_n$ are polynomials, we get the
following statement: If the Jacobian determinant associated to a system of $n$ polynomial equations in $n$ variables with complex coefficients is identically zero, then the system has no isolated solutions.