Polynomials and the exponent of matrix multiplication

Abstract

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

14 pages + appendix of 3 pages with numerical decompositions

References in corpus (2)

• On generic identifiability of symmetric tensors of subgeneric rank
• A $2n^2-log(n)-1$ lower bound for the border rank of matrix multiplication

Cited by in corpus (3)

• On the geometry of geometric rank
• Restricted Secant Varieties of Grassmannians
• Waring decompositions of the product of two quadrics: the small rank cases