Equations for lower bounds on border rank

Understand

We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r.

• We apply these methods to several cases including the case r = 6 in the space of bilinear maps C^4 x C^4 -> C^4.
• This space of bilinear maps includes the matrix multiplication operator M_2 for two by two matrices.
• We show these newly obtained polynomials do not vanish on the matrix multiplication operator M_2, which gives a new proof that the border rank of the multiplication of 2 x 2 matrices is seven.