  We consider systems of Laurent polynomials with support on a fixed point
configuration. In the non-defective case, the closure of the locus of
coefficients giving a non-degenerate multiple root of the system is defined by
a polynomial called the mixed discriminant. We define a related polynomial
called the multivariate iterated discriminant, generalizing the classical
Sch\"afli method for hyperdeterminants. This iterated discriminant is easier to
compute and we prove that it is always divisible by the mixed discriminant. We
show that tangent intersections can be computed via iteration if and only if
the singular locus of a corresponding dual variety has sufficiently high
codimension. We also study when point configurations corresponding to
Segre-Veronese varieties and to the lattice points of planar smooth polygons,
have their iterated discriminant equal to their mixed discriminant.
