Rank and optimal computation of generic tensors

Dedicated to Alexander M. Ostrowski on the occasion of his ninetieth birthday.
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Abstract

The typical rank (= maximal border rank) of tensors of a given size and the set of optimal bilinear computations of typical tensors of a given rank are investigated. For the size (n, n, 3) with n odd, the complement of the set of tensors of maximal border rank is a hypersurface. Its equation is given.

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This work was supported in part through funds provided by the National Science Foundation under Grant No. MCS 7412997 through a subcontract from MIT to the University of Washington.