Border Rank Is Not Multiplicative under the Tensor Product
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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
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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