Corrigendum to: “On the cactus rank of cubic forms”. (Corrigendum to: “On the cactus rank of cubics forms”.) (English) Zbl 07923132
From the text: The main results in the authors’ paper [ibid. 50, 291–297 (2013; Zbl 1258.14063)] concern minimal length apolar subschemes to cubic forms. Apolarity is defined via an action on polynomials by a dual polynomial ring. In the paper
the action is defined as differentiation: \(T := \mathbb{C}[y_0,\dots, y_n]\) acting on \(S = \mathbb{C}[x_0,\dots, x_n]\) by
\[
y_i(x_i)=\frac{d}{dx_j}(x_i)=\delta_{ij}.
\]
Which means that \(y^2_j (x^2_i) = 2\delta_{ij}\). Alternatively one may define the action by contraction, which means that \(y^2_j (x^2_i) = \delta_{ij}\). With the action defined by differentiation, the proof of Lemma 2 is incorrect, however with the action defined by contraction, the proof is correct. Differentiation should therefore be substituted by contraction in the paper, since this lemma is used throughout.
MSC:
14N05 | Projective techniques in algebraic geometry |
15A69 | Multilinear algebra, tensor calculus |
13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
Citations:
Zbl 1258.14063References:
[1] | Bernardi, Alessandra; Ranestad, Kristian, On the cactus rank of cubics forms, J. Symb. Comput., 50, 291-297, 2013 · Zbl 1258.14063 |
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