Mathematics – Algebraic Geometry
Scientific paper
2011-10-18
Mathematics
Algebraic Geometry
34 pages, 3 figures
Scientific paper
What polynomial in the coefficients of a system of algebraic equations should be called its discriminant? We prove a package of facts that provide a possible answer. Let us call a system typical, if the homeomorphic type of its set of solutions does not change as we perturb its (non-zero) coefficients. The set of all atypical systems turns out to be a hypersurface in the space of all systems of k equations in n variables, whose monomials are contained in k given finite sets. The hypersurface B contains all systems that have a singular solution, this stratum is conventionally called the discriminant, and the codimension of its components has not been fully understood yet (e.g. dual defect polytopes are not classified), so the purity of dimension of B looks somewhat surprising. We deduce it from a certain tropical purity fact of independent interest. A generic system of equations in a component B_i of the hypersurface B differs from a typical system by the Euler characteristic of its set of solutions. Regarding the difference of these Euler characteristics as the multiplicity of B_i, we turn B into an effective divisor, whose equation we call the Euler discriminant by the following reasons. Firstly, it vanishes exactly at those systems that have a singular solution (possibly at infinity). Secondly, despite its topological definition, there is a linear-algebraic formula for it, and a positive formula for its Newton polytope. Thirdly, it interpolates many classical objects (sparse resultant, A-determinant, Teissier discriminant) and inherits many of their nice properties. This allows to specialize our results to generic polynomial maps: the bifurcation set of a nondegenerate dominant polynomial map is always a hypersurface, and a generic atypical fiber of such a map differs from a typical one by its Euler characteristic.
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