  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. This 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 toric
varieties are not classified), so the purity of dimension of B looks somewhat
surprising. We deduce it from a similar tropical purity fact.
  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 simple
linear-algebraic formula for it, and a positive formula for its Newton
polytope. Thirdly, it interpolates many classical objects (sparse resultant,
A-determinant, discriminant of deformation) and inherits many of their nice
properties. This allows to specialize our results to generic polynomial maps:
the bifurcation set of a dominant polynomial map, whose components are generic
linear combinations of finitely many monomials, is always a hypersurface, and a
generic atypical fiber of such a map differs from a typical one by its Euler
characteristic.
