Nontriviality of equations and explicit tensors in CmCmCm of border rank at least 2m2 Academic Article

abstract

• 2014 Elsevier B.V. For even (resp. odd) m, I show the Young-flattening equations for border rank of tensors in CmCmCm of [7] are nontrivial up to border rank 2m-3 (resp. 2m-5) by writing down explicit tensors on which the equations do not vanish. Thus these tensors have border rank at least 2m-2 (resp. 2m-4). The result implies that there are nontrivial equations for border rank 2n2-n that vanish on the matrix multiplication tensor for nn matrices. I also study the border rank of the tensors of [1] and the equations of [4]. I show the tensors T2kCkC2kC2k of [1], despite having rank equal to 2k+1-1, have border rank equal to 2k. I show the equations for border rank of [4] are trivial in the case of border rank 2m-1 and determine their precise non-vanishing on the matrix multiplication tensor.

authors

• Landsberg, Joseph

published proceedings

• Journal of Pure and Applied Algebra

author list (cited authors)

• Landsberg, J. M.

citation count

• 6

complete list of authors

• Landsberg, JM

publication date

• August 2015

publisher

• Elsevier  Publisher

published in

• Journal of Pure and Applied Algebra  Journal

Digital Object Identifier (DOI)

• 10.1016/j.jpaa.2014.12.016

start page

• 3677

end page

• 3684

volume

• 219

issue

• 8

URL

• http%3A%2F%2Fdx.doi.org%2F10.1016%2Fj.jpaa.2014.12.016