We consider the problem of determining the symmetric tensor rank for
symmetric tensors with an algebraic geometry approach. We give algorithms for
computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors
of small border rank. From a geometric point of view, we describe the symmetric
rank strata for some secant varieties of Veronese varieties.Comment: Journal of Symbolic Computatio
International audienceWe 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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