Iterated and mixed discriminants

• Published in: Journal of combinatorial algebra Vol. 7; no. 1; pp. 45 - 81
• Main Authors: Dickenstein, Alicia, Di Rocco, Sandra, Morrison, Ralph
• Format: Journal Article
• Language: English
• Published: 01.01.2023
• Subjects: algebraic varieties; discriminants; Polynomial systems
• ISSN: 2415-6302; 2415-6310; 2415-6310
• DOI: 10.4171/jca/68

Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schafli for the study of hyperde-terminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hyper -surfaces was given by Ottaviani. We propose and study a generalization of this question for 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 Schafli 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.