On the cactus rank of cubic forms

Alessandra Bernardi 1 Kristian Ranestad 2
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
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 $2n$.
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Alessandra Bernardi, Kristian Ranestad. On the cactus rank of cubic forms. Journal of Symbolic Computation, Elsevier, 2013, 50, pp.291-297. ⟨10.1016/j.jsc.2012.08.001⟩. ⟨inria-00630456v3⟩

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