Effective vanishing order of the Levi determinant

On a smooth domain in complex n space of finite D'Angelo q-type at a point, an effective upper bound for the vanishing order of the Levi determinant $\text{coeff}\{\partial r \wedge \dbar r \wedge (\partial \dbar r)^{n-q}\}$ at that point is given in terms of the D'Angelo q-type, the dimension of the space n, and q itself. The argument uses Catlin's notion of a boundary system as well as techniques pioneered by John D'Angelo.