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Equations for lower bounds on border rank
Citation key HIL-Equations-For-Lower-Bounds-On-Border-Rank
Author Jonathan David Hauenstein and Christian Ikenmeyer and Joseph Montague Landsberg
Pages 372–383
Year 2013
DOI 10.1080/10586458.2013.825892
Journal Experimental Mathematics
Volume 22
Number 4
Abstract 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 $\mathbb C^4 × \mathbb C^4 \to \mathbb C^4$. 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 $2 × 2$ matrices is seven. Other examples are considered along with an explanation of how to implement the methods.
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