The discriminant of a system of equations

Abstract

We introduce and study the discriminant of a system of polynomial equations with indeterminate coefficients, generalizing a number of known special cases, such as the sparse resultant and the A-determinant, and inheriting their nice properties. One such property is the fact that the discriminant is always a hypersurface in the space of systems of equations (which seems unexpected when compared to the existence of dual defect toric varieties). This fact reflects the combinatorics of a certain partial order relation on the set of tuples of faces of a tuple of polytopes (the Newton polytopes of the equations in our case), generalizing the poset of faces of one polytope. This combinatorics is the main challenge in our work, and can be reduced to a certain fact of tropical geometry, which resembles the geometry of the discriminants that we study. As a sample application of our version of the multivariate discriminant, we prove that atypical fibers of a generic polynomial map are distinguished by their Euler characteristics, and the bifurcation set of such a map is a hypersurface, whose degree we explicitly compute.