Discriminant locus of a system of $n$ Laurent polynomials in $n$ variables

We deal with a system of $n$ algebraic equations in $n$ variables. Assume that all sets of exponents of the monomials are fixed, while all coefficients are variable. We study the discriminant locus which is the closure of the set of coefficients for which the system has multiple roots with non-zero coordinates. We present a parametrization for the irreducible components of the discriminants locus, depending on the coefficients of all equations. In the case of a 1-codimensional component, the parametrization is the inverse to the logarithmic Gauss mapping (an analog of Kapranov's theorem for the A-discriminant). Our research is based on the linearization of the algebraic system and on the parametrization of the linearization critical values set. This is joint work with August Tsikh.