On the cactus rank of cubics forms

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 $n\ge 8$, 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 $2n$.

We thank Joachim Jelisiejew for pointing out that Lemma 2 is correct as stated only when using apolarity by contraction (as reflected in the current version of this paper, updated in May 2024) instead of apolarity by differentiation (as it was in the originally published version)

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• Computing symmetric rank for symmetric tensors
• Eigenvectors of tensors and algorithms for Waring decomposition
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• Equations for secant varieties to Veronese varieties
• Gonality, apolarity and hypercubics
• Fermat hypersurfaces and Subcanonical curves
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• Border Ranks of Monomials
• A note on the cactus rank for Segre-Veronese varieties
• Symmetric Decomposition of the Associated Graded Algebra of an Artinian Gorenstein Algebra
• On schemes evinced by generalized additive decompositions and their regularity
• Inverse Systems of Zero-dimensional Schemes in P^n
• A comparison of different notions of ranks of symmetric tensors
• Waring and cactus ranks and Strong Lefschetz Property for annihilators of symmetric forms
• Geometric conditions for strict submultiplicativity of rank and border rank
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• Border Waring Rank via Asymptotic Rank
• Apolarity for determinants and permanents of generic symmetric matrices
• The stratification by rank for homogeneous polynomials with border rank 5 which essentially depend on 5 variables
• Waring decompositions of the product of two quadrics: the small rank cases
• Some examples of forms of high rank