Border Rank Is Not Multiplicative under the Tensor Product

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It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper bounds were obtained with the help of border rank. It was left open whether border rank itself can be strictly submultiplicative. We answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop. Our results also imply strict submultiplicativity for cactus rank and border cactus rank.
Original languageEnglish
JournalSIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY
Volume3
Issue number2
Pages (from-to)231–255
ISSN2470-6566
DOIs
Publication statusPublished - 2019

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