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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Decomposable tensors as a quadratic variety
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by Robert Grone PDF
Proc. Amer. Math. Soc. 64 (1977), 227-230 Request permission


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.
  • Marvin Marcus, Finite dimensional multilinear algebra. Part II, Pure and Applied Mathematics, Vol. 23, Marcel Dekker, Inc., New York, 1975. MR 0401796
  • Marvin Marcus, A dimension inequality for multilinear functions, Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), Academic Press, New York, 1972, pp. 217–224. MR 0332849
  • William Watkins, Linear maps and tensor rank, J. Algebra 38 (1976), no. 1, 75–84. MR 424855, DOI 10.1016/0021-8693(76)90244-1
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 227-230
  • MSC: Primary 14M15; Secondary 15A69
  • DOI:
  • MathSciNet review: 0472853