“Iterated sparse discriminants and singular intersections of hypersurfaces”
It is well known that two generic quadric surfaces intersect in a nonsingular quartic space curve, but when the intersection is not transverse this intersection curve may degenerate to a finite number of different possible types of singular curves. In the nice paper by Farouki et al. (1989), the authors formulate a way of computing the condition for a degenerate intersection in this case, which refines in the real case and with an algorithmic point of view a classical treatise by Bromwich (1906). Independently, Schläfli (1953) studied the degenerate intersection of two hypersurfaces described by multilinear equations.
In joint work in progress with S. di Rocco and R. Morrison, Prof. Dickenstein presents a general framework of iterated sparse discriminants to characterize the singular intersection of hypersurfaces with a given monomial support A, which generalizes both previous situations. They study the connection of iterated discriminants with the notion of mixed discriminant and the singularities of the sparse discriminant associated with A.