Apolarity, steampunk canonical forms and the obvious inner product -- April remix

There is an obvious and very old inner product on the vector space of complex forms of degree d in n variables, under which apolarity becomes orthogonality. We combine this with some other very old techniques to find some exotic canonical forms, especially for binary forms and forms of low degree. The methods are elementary, if not actually simple-minded. A very preliminary version of this material was given in this seminar in February. This presentation will reflect work done since and will NOT assume that the audience saw the first talk.