The discriminant locus of a system of n Laurent polynomials in n variables

摘要

We consider a system of n algebraic equations in n variables, where the exponents of the monomials in each equation are fixed while all the coefficients vary. The discriminant locus of such a system is the closure of the set of all coefficients for which the system has multiple roots with non-zero coordinates. For dehomogenized discriminant loci, we give parametrizations of those irreducible components that depend on the coefficients of all the equations. We prove that if such a component has codimension 1, then the parametrization is inverse to the logarithmic Gauss map of the component (an analogue of Kapranov's result for the A-discriminant). Our argument is based on the linearization of algebraic systems and the parametrization of the set of its critical values.