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Abstract
We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds for the border rank of a given tensor. Landsberg used such techniques to give non-trivial equations for the tensors of border rank at most in
if m is even. He also gave such equations for tensors of border rank at most
in
if m is odd. Using concavity of tensor blow-ups we show non-trivial equations for tensors of border rank
in
for odd m for any field
of characteristic 0. We also give another proof of the regularity lemma by Ivanyos, Qiao and Subrahmanyam.
Notes
No potential conflict of interest was reported by the authors.