Some new canonical forms for polynomials

Abstract

We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can be written uniquely as a sum of the cubes of linear forms $\ell_{ij}(x_i,...,x_j)$, $1 \le i \le j \le n$. A general ternary quartic form is the sum of the square of a quadratic form and three fourth powers of linear forms. The methods are classical and elementary.