On the cactus rank of cubics forms

Abstract

We prove that the smallest degree of an apolar 0-dimensional scheme of a general cubic form in $n+1$ variables is at most $2n+2$, when $n\geq 8$, and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is $n+2$, while the rank is at least $2n$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Example 1
• Conjecture 1