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.

I have spoken about this material under the title "steampunk canonical forms". This is the final revised version which has been accepted by the Pacific Journal of Mathematics. Apart from the usual improvements which come after a thoughtful refereeing, Theorem 1.8 is new

References in corpus (2)

• On the Waring problem for polynomial rings
• The Waring rank of the sum of pairwise coprime monomials

Cited by in corpus (6)

• On generic and maximal k-ranks of binary forms
• Algebraic stories from one and from the other pockets
• Linearly dependent powers of binary quadratic forms
• Semialgebraic decomposition of real binary forms of a given degree's space
• On real Waring decompositions of real binary forms
• An operational construction of the sum of two non-commuting observables in quantum theory and related constructions