A math friend pointed out the following (slightly edited): ----- The roots-to-coefficients map, taking (r1,r2,...,rn) to the vector of elementary symmetric functions in the ri, (\sigma1,\sigma2,...,\sigma_n), has the square root of the discriminant as the determinant of its derivative/Jacobian matrix. [This may be a little imprecise.] Also, two triangles whose vertices are viewed as complex numbers, (z1,z2,z3) and (w1,w2,w3), are similar with zj going to wj for j=1,2,3 iff \det(1 1 1 \\ z1 z2 z3 \\ w1 w2 w3) = 0. Especially, taking w1=z2, w2=z3, w3=z1 gives a polynomial condition for (z1,z2,z3) to be equilateral. ----- —Dan
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?