The beautiful world of A-discriminants

Alicia Dickenstein (Universidad de Buenos Aires)

This talk will be a gentle introduction to the theory of A-discriminants, a far-going generalization of the discriminant of a univariate polynomial proposed by Gelfand, Kapranov and Zelevinsky in the late 80's, who also gave many of their combinatorial properties.

Given a finite set A of lattice points in n variables, the associated A-discriminant is an irreducible polynomial which vanishes on a vector of coefficients $(x_a : a \in A)$ whenever the hypersurface of the n-torus with equation $\sum_{a \in A} x_a t^a =0$ has a singular point.

Sparse resultants are instances of A-discriminants, and conversely, A-discriminants turn out to be the principal factor of certain specializations of sparse resultants.

Discriminants appear in computer vision, singularities of differential equations, numerical unstability of algebraic equations, singularities of mechanisms, number theory, etc.

We will focus on the computation of A-discriminants associated to codimension two configurations, giving in this case a precise description of its Newton polytope and its degree (joint work with Bernd Sturmfels). We'll also discuss several open problems.