Equations for Lower Bounds on Border Rank

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 . This space of bilinear maps includes the matrix multiplication operator M ₂ for 2×2 matrices. We show that 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.

2000 AMS Subject Classification:

• 68Q17

Keywords:

• border rank
• matrix multiplication

Acknowledgments

We thank Peter Bürgisser for important discussions and suggestions, and the anonymous reviewer for many helpful comments.

Hauenstein's research was supported in part by AFOSR grant FA8650-13-1-7317 and NSF grant DMS-1262428. Ikenmeyer's research was supported in part by DFG grant BU 1371/3-2. Landsberg's research was supported in part by NSF grant DMS-1006353.