Theory of Computing
-------------------
Title     : New Lower Bounds for the Border Rank of Matrix Multiplication
Authors   : Joseph M. Landsberg and Giorgio Ottaviani
Volume    : 11
Number    : 11
Pages     : 285-298
URL       : https://theoryofcomputing.org/articles/v011a011

Abstract
--------

  The border rank of the matrix multiplication operator for
  $n\times n$ matrices is a standard measure of its
  complexity. Using techniques from algebraic geometry and
  representation theory, we show the border rank is at least
  $2n^2-n$.  Our bounds are better than the previous lower bound
  (due to Lickteig in 1985) of $3n^2/2 + n/2 - 1$ for
  all $n\geq 3$.  The bounds are obtained by finding new equations
  that bilinear maps of small border rank must satisfy, i.e., new
  equations for secant varieties of triple Segre products, that matrix
  multiplication fails to satisfy.