Nontriviality of equations and explicit tensors in Cm⊗Cm⊗Cm of border rank at least 2m−2 Academic Article uri icon

abstract

  • © 2014 Elsevier B.V. For even (resp. odd) m, I show the Young-flattening equations for border rank of tensors in Cm⊗Cm⊗Cm 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 n×n matrices. I also study the border rank of the tensors of [1] and the equations of [4]. I show the tensors T2k∈Ck⊗C2k⊗C2k 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.

author list (cited authors)

  • Landsberg, J. M.

citation count

  • 3

publication date

  • August 2015