Border Rank Is Not Multiplicative under the Tensor Product

Matthias Christandl; Fulvio Gesmundo; Asger Kjærulff Jensen

doi:10.1137/18M1174829

Border Rank Is Not Multiplicative under the Tensor Product

Matthias Christandl, Fulvio Gesmundo, Asger Kjærulff Jensen

Department of Mathematical Sciences

4 Citations (Scopus)

Abstract

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 language	English
Journal	SIAM Journal on Applied Algebra and Geometry
Volume	3
Issue number	2
Pages (from-to)	231–255
ISSN	2470-6566
DOIs	https://doi.org/10.1137/18M1174829
Publication status	Published - 2019

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10.1137/18M1174829

Cite this

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AU - Jensen, Asger Kjærulff

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DO - 10.1137/18M1174829

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