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# Decomposable Tensors as a Quadratic Variety

Robert Grone
Proceedings of the American Mathematical Society
Vol. 64, No. 2 (Jun., 1977), pp. 227-230
DOI: 10.2307/2041432
Stable URL: http://www.jstor.org/stable/2041432
Page Count: 4
Let $V_i$ be a finite dimensional vector space over a field $F$ for each $i = 1, 2, \ldots, m$, and let $z$ be a tensor in $V_1 \otimes \cdots \otimes V_m$. In this paper a set of homogeneous quadratic polynomials in the coordinates of $z$ is exhibited for which the associated variety is the set of decomposable tensors. In addition, a question concerning the maximal tensor rank in such a situation is answered, and an application to other symmetry classes of tensors is cited.