Iterated and mixed discriminants

TL;DR

This paper introduces the multivariate iterated discriminant for systems of Laurent polynomials, showing its relation to the mixed discriminant and providing methods for computation and analysis of singularities.

Contribution

It defines the iterated discriminant as a generalization of classical hyperdeterminants, proves its divisibility by the mixed discriminant, and explores conditions for tangent intersection computations.

Findings

Iterated discriminant is easier to compute than the mixed discriminant.

Iterated discriminant is always divisible by the mixed discriminant.

Conditions identified when iterated and mixed discriminants coincide for specific configurations.

Abstract

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.