From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Geometric significance of algebraic discriminant ?

I recall learning a bunch of algebra about the discriminant, which becomes zero when there are coincident roots.

https://en.wikipedia.org/wiki/Discriminant

Has someone come up with geometric insights about this particular formula ?

In the case of a quadratic, the formula is (x1-x2)^2, but this isn't the real number |x1-x2|^2. Perhaps the norm of the discriminant (DD*) is more important?

What about the discriminant of the cubic ? Shouldn't this say something interesting about the triangle in the complex plane formed by the roots?

Perhaps one might get some inspiration through the book Gelfand, Kapranov, Zelevinsky: Discriminants, resultants, and multidimensional determinants (1994), may be already in Chap 1 Though I admit that I missed the energy to 'read' it