Polynomials and the exponent of matrix multiplication

Abstract

We define tensors, corresponding to cubic polynomials, which have the same exponent $ω$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix $A$ by $sM_n(A)=trace(A^3)$. The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent $ω$.

14 pages + appendix of 3 pages with numerical decompositions