Quartic discriminants and tensor invariants
J.F. Blinn · IEEE Computer Graphics and Applications · 2002
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 /spl Delta//sub 4/ of a homogeneous quartic f(x,w) = Ax/sup 4/+4Bx/sup 3/w+6Cx/sup 2/w/sup 2/+4Dxw/sup 3/+Ew/sup 4/ = 0 is /spl Delta//sub 4/ = 27(I/sub 3/)/sup 2/-(I/sub 2/)/sup 3/, where I/sub 2/ = AE-4BD+3C/sup 2/ and I/sub 3/ = ACE-AD/sup 2/-B/sup 2/E+2BCD-C/sup 3/ (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.