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2012

Explicit tensors of border rank at least 2n-1

Landsberg, J. M.

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For odd n, I write down tensors in C^n\otimes C^n\otimes C^n of border rank 2n-1, showing the non-triviality of the Young-flattening equations of Landsberg-Ottaviani.

I also study the border rank of the tensors of Alexeev et.
al., showing the tensors their tensors T_{2^k}, despite having rank equal to 2^{k+1}-1, have border rank equal to 2^k, the minimum of any concise tensor.
I also study the equations of Griesser.

Explicit tensors of border rank at least 2n-1 · Around

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V. Strassen, Rank and optimal computation of generic tensors , Linear Algebra Appl. 52/53
1983
Earlier work this paper cites.
Thomas Lickteig, A note on border rank , Inform. Process. Lett. 18
1984
Earlier work this paper cites.
by same author, Typical tensorial rank , Linear Algebra Appl. 69
1985
Earlier work this paper cites.

Similar

B. Griesser, A lower bound for the border rank of a bilinear map , Calcolo 23
1987
Cited alongside, same era.
Peter Bürgisser, Michael Clausen, and M. Amin Shokrollahi, Algebraic complexity theory , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 315, Springer-Verlag, Berlin, 1997, With the collaboration of Thomas Lickteig. MR 99c:68002
1997
Cited alongside, same era.
J. M. Landsberg, New lower bounds for the rank of matrix multiplication , preprint arXiv:1206.1530
Original
Cited in the paper.
J.M. Landsberg and Giorgio Ottaviani, Equations for secant varieties of veronese and other varieties , arXiv:1111.4567, to appear in Annali di Matematica Pura e Applicata
Original
Cited in the paper.
by same author, New lower bounds for the border rank of matrix multiplication , preprint, arXiv:1112.6007
Original
Cited in the paper.

Then

Boris Alexeev, Michael Forbes, and Jacob Tsimerman, Tensor rank: some lower and upper bounds , IEEE Conference on Computational Complexity, IEEE Computer Society, Feb 2011, pp. 283–291
2011
Later among the works it cites.
B. Weitz, An improvement on rank of explicit tensors , arXiv 1102.0580 (2011)
2011
Later among the works it cites.

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