On the cactus rank of cubics forms

TL;DR

This paper investigates the minimal degree of apolar schemes for general cubic forms, establishing bounds that relate to the forms' rank and revealing differences between reducible and irreducible cases.

Contribution

It provides new bounds on the minimal degree of apolar schemes for general cubic forms, highlighting differences based on reducibility and advancing understanding of cubic form ranks.

Findings

Minimal degree of apolar schemes for general cubics is at most 2n+2 for n≥8.

For reducible cubics, the minimal apolar scheme degree is n+2.

The rank of general cubics exceeds the minimal degree of their apolar schemes.

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 $2n$.