Typical tensorial rank

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Abstract

Upper bounds on the typical rank R(n, m, l) of tensors ( = maximal border rank = rank of almost all tensors) of a given shape (n, m, l) are presented. These improve previous results by A tkinson and Lloyd. For cubic shape tensors the typical rank is determined exactly: R(n, n, n) = ⌈ n3/(3n − 2) ⌉ (n ≠ 3)

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The results of this paper were presented at the Oberwolfach Conference on Komplexitätstheorie, November 1981.