On noncommutative rank and tensor rank
Abstract
We study the relationship between the commutative and the noncommutative 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 nontrivial equations for the tensors of border rank at most $2m3$ in $K^m\otimes K^m\otimes K^m$ if $m$ is even. He also gave such equations for tensors of border rank at most $2m5$ in $K^m\otimes K^m\otimes K^m$ if $m$ is odd. Using concavity of tensor blowups we show nontrivial equations for tensors of border rank $2m4$ in $K^m \otimes K^m \otimes K^m$ for odd $m$ for any field $K$ of characteristic 0. We also give another proof of the regularity lemma by Ivanyos, Qiao and Subrahmanyam.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.06701
 Bibcode:
 2016arXiv160606701D
 Keywords:

 Mathematics  Rings and Algebras;
 16R50;
 17A35;
 15A69
 EPrint:
 15 pages