Geodesic
The discriminant locus of a~system of $n$ Laurent polynomials in $n$ variables
Izvestiya. Mathematics , Tome 76 (2012) no. 5, pp. 881-906

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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.
Keywords: linearization of an algebraic system
Mots-clés : discriminant locus, logarithmic Gauss map. 
@article{IM2_2012_76_5_a1,
     author = {I. A. Antipova and A. K. Tsikh},
     title = {The discriminant locus of a~system of $n$ {Laurent} polynomials in $n$ variables},
     journal = {Izvestiya. Mathematics },
     pages = {881--906},
     publisher = {mathdoc},
     volume = {76},
     number = {5},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2012_76_5_a1/}
}
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I. A. Antipova; A. K. Tsikh. The discriminant locus of a~system of $n$ Laurent polynomials in $n$ variables. Izvestiya. Mathematics , Tome 76 (2012) no. 5, pp. 881-906. http://geodesic.mathdoc.fr/item/IM2_2012_76_5_a1/