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) = ax^{2} + 2bx + c. This corresponds to a matrix

M = | ( | a | b | ) |

b | c |

The discriminant is simply the negative determinant of the matrix

Δ_{2} = b^{2 }– ac

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

a_{111} = a

a_{211} = a_{121} = a_{112} = b

a_{221} = a_{212} = a_{122}= c

a_{222} = d

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

Δ_{3} = 3b^{2}c^{2 }+ 6abcd – 4b^{3}d – 4ac^{3 }–^{ }a^{2}d^{2}

Next up is the quartic P(x) = ax^{4} + 4bx^{3}+ 6cx^{2} + 4dx + e. This corresponds to a 2x2x2x2 tensor whose components are

a_{1111} = a

a_{2111} = a_{1211} = a_{1121} = a_{1112} = b

a_{2211} = a_{2121} = a_{1221}= a_{2112} = a_{1212} = a_{1122}= c

a_{2221} = a_{2212} = a_{2122} = a_{1222} = d

a_{2222} = e

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

X = ε^{im }ε^{jn }ε^{kp }ε^{lq }a_{ijkl }a_{mnpq}

This corresponds to a polynomial invariant

g_{2} = ae – 4bd + 3c^{2}

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

g_{3} = |
a | b | c | ||

b | c | d | |||

c | d | e |

The discriminant is a dependent of these two invariants given by

Δ_{4 }= g_{2}^{3} – 27g_{3}^{2}

October 11, 2008 at 10:48 am

[…] the ring of invariants on the 2×2×2×2 hypermatrix. We know that because we saw that the discriminant of the quartic can be written in terms of two simpler Sl(2) invariants Δ4 = g23 – 27g32 . These invariants also […]