Quartic Discriminants

October 9, 2008

In the last post we discussed how the discriminant for a polynomial of degree n can be seen as an SL(2) invariants on a fully symmetric tensor of rank n. We have also seen how invariants of hypermatrix tensors can be constructed using alternating forms. Of course a similar mathod can be applied for constructing polynomial invariants. The only difference is that e can contract the alternating tensors over the form indices in any order since they are all transformed simualtaneously.

I’ll walk through this for the lowest order polynomials. First a warning, I am changing notation for the coefficients to avoid fractions and to be consistent with the most common usage.

The quadratic is now written P(x) = ax2 + 2bx + c. This corresponds to a matrix

 M = ( a b ) b c

The discriminant is simply the negative determinant of the matrix

Δ2 = b2 – ac

Next we looked at the cubic which I will now right as P(x) = ax3 + 3bx2+ 3cx + d. This corresponds to a 2x2x2 tensor with components

a111 = a

a211 = a121 = a112 = b

a221 = a212 = a122= c

a222 = d

The only independent invariant is the discriminant which comes from Cayley’s hyperdeterminant

Δ3 = 3b2c2 + 6abcd – 4b3d – 4ac3 a2d2

Next up is the quartic P(x) = ax4 + 4bx3+ 6cx2 + 4dx + e. This corresponds to a 2x2x2x2 tensor whose components are

a1111 = a

a2111 = a1211 = a1121 = a1112 = b

a2211 = a2121 = a1221= a2112 = a1212 = a1122= c

a2221 = a2212 = a2122 = a1222 = d

a2222 = e

This has a quadratic invariant which can be written in terms of alternating tensors like this

X =   εim εjn εkp εlq aijkl amnpq

This corresponds to a polynomial invariant

g2 = ae – 4bd + 3c2

There is another cubic invariant which is most easily written as a determinant

 g3 = a b c b c d c d e

The discriminant is a dependent of these two invariants given by

Δ4 = g23 – 27g32