  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 $sM_n$ 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 $\omega$.
