Polynomial discriminants. I. Matrix magic

I like beautiful equations, but beauty is sometimes subtle or hidden by bad notation. I reveal some of the hidden beauty in the explicit formulation of the discriminants of polynomials. Along the way I drag in some clever algebra, promote some notational schemes from mathematical physics and illustrate some ways of visualizing homogeneous space. This will ultimately lead us to some interesting ways to find roots of these polynomials, a task that will become more and more important as we computer graphicists struggle to break free of the tyranny of the polygon and move into rendering higher order surfaces. First I review discriminants.