Quartic discriminants and tensor invariants

Discusses the calculation of discriminants of polynomials. The discriminant is a function of the coefficients that indicates if the polynomial has any double roots. The discriminant Δ₄ of a homogeneous quartic f(x,w) = Ax⁴+4Bx³w+6Cx² w²+4Dxw³+Ew⁴ = 0 is Δ₄ = 27(I₃)²-(I₂)³, where I₂ = AE-4BD+3C² and I₃ = ACE-AD ²-B²E+2BCD-C³ (this is the Hilbert representation). The author shows how to write the discriminant as a tensor diagram. The discriminant of a polynomial is an example of an invariant quantity. Tensor diagrams are particularly useful to express invariant quantities. Adding a dimension moves us from the world of (1D) homogeneous polynomials to 2D homogeneous (2DH) geometry (curves in the projective plane). It is shown that a relationship exists between the possible root structures of a 4th-order polynomial and the possible degeneracies of a 3rd-order curve