Some new canonical forms for polynomials

TL;DR

This paper introduces new canonical forms for polynomials over complex numbers, providing explicit representations for binary quartics, cubic forms, and ternary quartics, using classical and elementary methods.

Contribution

It presents novel canonical representations for various polynomial forms, including binary quartics, cubic forms, and ternary quartics, with unique decompositions.

Findings

Binary quartics as quadratic squares plus linear fourth powers

Unique sum-of-cubes representation for cubic forms in multiple variables

Ternary quartics as quadratic plus three linear fourth powers

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.