User:Hellohihihihi/sandbox

Discriminant of a differentiable function
In differential topology, the discriminant of a differentiable function f is the image of the critical locus under f. In particular, it is a set.

OR: image of the pts where not a submersion?

This notion of discriminant is related to the classical discriminant of a polynomial. For instance, let Δ be the classical discriminant of the cubic polynomial
 * $$ F(x,a,b,c,d)=ax^3+bx^2+cx+d.$$

If $$p:\mathbb{R}^5\to \mathbb{R}^4$$ is the projection map
 * $$ p(x,a,b,c,d)=(a,b,c,d),$$

and $$p|_{\{F=0\}}$$ denotes the restriction of $$p$$ to the hypersurface defined by $$F$$, then the discriminant of $$p|_{\{F=0\}}$$ is the hypersurface in $$\mathbb{R}^4$$ defined by Δ.

In this example, $$F$$ is an unfolding of $$f(x)=x^3$$, with $$p|_{\{F=0\}}$$ constructed as the restriction of the projection onto the parameter space of the unfolding.