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 >= 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.
On the cactus rank of cubic forms
BERNARDI, Alessandra;
2013-01-01
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 >= 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.
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