  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. Other
examples are considered along with an explanation of how to implement the
methods.
