Suppose the polynomial is monic and irreducible, and has a root x. Then the field K := Q[x], viewed as a vector space over Q, has a basis of {1, x, x^2, ..., x^(n-1)}. We can endow the vector space with an inner product, namely (y, z) = tr(yz*), which gives rise to a natural metric on the field (which does not dependent on the choice of x used to generate K). Then the discriminant is the squared volume of the parallelepiped generated by {1, x, x^2, ..., x^(n-1)}. Sincerely, Adam P. Goucher
Sent: Wednesday, July 01, 2015 at 9:19 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Geometric significance of algebraic discriminant ?
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?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun