The discriminant is also the square of the Vandermonde determinant | 1 1 ... 1 || x1 x2 ... xn || x1^2 x2^2 ... xn^2 || ... | | x1^(n-1) x2^(n-1) ... xn^(n-1) | Perhaps you can find some interpretation of the determinant as a volume. -- Gene From: Henry Baker <hbaker1@pipeline.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, June 30, 2015 5:06 PM Subject: Re: [math-fun] Geometric significance of algebraic discriminant ? What Dan says is all true, but I'm looking for some geometric significance to the number calculated by the discriminant. Is it a volume in some hyperspace? What is the significance of its magnitude? Its phase? At 03:35 PM 6/30/2015, Dan Asimov wrote:
On Jun 30, 2015, at 3:05 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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?
The discriminant disc(P) of any polynomial P(x) in K[x] (K being some field) is the product of the squares of the differences between all pairs of roots.
(When K is a subfield of the reals, disc(P) is always real, because it's a symmetric polynomial Q(z_1,...,z_n) of the roots z_1,...,z_n of P(x), and it's a theorem that all symmetric polynomials are polynomials in the elementary symmetric functions . . . and the the elementary symmetric functions of the roots of P(x) are of course the coefficients.)