The cactus rank of cubic forms

Bernardi, Alessandra ; Ranestad, Kristian (2011) The cactus rank of cubic forms. [Preprint]

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Abstract

We prove that the smallest degree of an apolar $0$-dimensional scheme to 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. When $n=8$ we show that the bound is sharp, i.e. the smallest degree of an apolar subscheme is $18$.

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