• Corpus ID: 221878835

# Bad and good news for Strassen's laser method: Border rank of the 3x3 permanent and strict submultiplicativity

@article{Conner2020BadAG,
title={Bad and good news for Strassen's laser method: Border rank of the 3x3 permanent and strict submultiplicativity},
author={Austin Conner and Hang Huang and Joseph Landsberg},
journal={ArXiv},
year={2020},
volume={abs/2009.11391}
}
• Published 23 September 2020
• Mathematics
• ArXiv
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent $\omega$ of matrix multiplication. The Kronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the $3\times 3$ permanent, and could potentially be used to show $\omega=2$. We prove the negative result for complexity theory that its border rank is $16$, resolving a longstanding problem. Regarding its $q=4$ skew cousin in $C^5\otimes C^5\otimes C^5$, which could…
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