[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?
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.) —Dan
The Wikipedia article implies that Dan left out a factor of some power of the leading coefficient. I think I trust Dan on this one. On Tue, Jun 30, 2015 at 6:35 PM, Dan Asimov <dasimov@earthlink.net> 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.)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Actually I left out a factor of +-1 in the last independent clause, depending on the parity of the index of the coefficient. —Dan
On Jun 30, 2015, at 3:40 PM, Allan Wechsler <acwacw@gmail.com> wrote:
The Wikipedia article implies that Dan left out a factor of some power of the leading coefficient. I think I trust Dan on this one.
On Tue, Jun 30, 2015 at 6:35 PM, Dan Asimov <dasimov@earthlink.net> 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.)
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.)
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.)
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?
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
On 2015-07-01 15:09, Adam P. Goucher wrote:
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
Aha. Scrooge's accountants probably use square gallons instead of cubic acres. --rwg
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (7)
-
Adam P. Goucher -
Allan Wechsler -
Dan Asimov -
Dan Asimov -
Eugene Salamin -
Henry Baker -
rwg