Border Rank Is Not Multiplicative under the Tensor Product

Research output: Contribution to journal › Journal article › Research › peer-review

Standard

Border Rank Is Not Multiplicative under the Tensor Product. / Christandl, Matthias; Gesmundo, Fulvio; Jensen, Asger Kjærulff.

In: SIAM Journal on Applied Algebra and Geometry, Vol. 3, No. 2, 2019, p. 231–255.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Christandl, M, Gesmundo, F & Jensen, AK 2019, 'Border Rank Is Not Multiplicative under the Tensor Product', SIAM Journal on Applied Algebra and Geometry, vol. 3, no. 2, pp. 231–255. https://doi.org/10.1137/18M1174829

APA

Christandl, M., Gesmundo, F., & Jensen, A. K. (2019). Border Rank Is Not Multiplicative under the Tensor Product. SIAM Journal on Applied Algebra and Geometry, 3(2), 231–255. https://doi.org/10.1137/18M1174829

Vancouver

Christandl M, Gesmundo F, Jensen AK. Border Rank Is Not Multiplicative under the Tensor Product. SIAM Journal on Applied Algebra and Geometry. 2019;3(2):231–255. https://doi.org/10.1137/18M1174829

Author

Christandl, Matthias ; Gesmundo, Fulvio ; Jensen, Asger Kjærulff. / Border Rank Is Not Multiplicative under the Tensor Product. In: SIAM Journal on Applied Algebra and Geometry. 2019 ; Vol. 3, No. 2. pp. 231–255.

Bibtex

@article{3abb8dbb820e4888b0341cf6fba8a6e0,
title = "Border Rank Is Not Multiplicative under the Tensor Product",
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.",
author = "Matthias Christandl and Fulvio Gesmundo and Jensen, {Asger Kj{\ae}rulff}",
year = "2019",
doi = "10.1137/18M1174829",
language = "English",
volume = "3",
pages = "231–255",
journal = "SIAM Journal on Applied Algebra and Geometry",
issn = "2470-6566",
publisher = "Society for Industrial and Applied Mathematics",
number = "2",

}

RIS

TY - JOUR

T1 - Border Rank Is Not Multiplicative under the Tensor Product

AU - Christandl, Matthias

AU - Gesmundo, Fulvio

AU - Jensen, Asger Kjærulff

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

U2 - 10.1137/18M1174829

DO - 10.1137/18M1174829

M3 - Journal article

VL - 3

SP - 231

EP - 255

JO - SIAM Journal on Applied Algebra and Geometry

JF - SIAM Journal on Applied Algebra and Geometry

SN - 2470-6566

IS - 2

ER -

ID: 230844066