Polynomials and the exponent of matrix multiplication

The exponent of matrix multiplication is the smallest constant ω such that two n × n matrices may be multiplied by performing O(n ω+e ) arithmetic operations for every e > 0. Determining the constant ω is a central question in both computer science and mathematics. We define certain symmetric tensors, that is, cubic polynomials, and our main result is that their symmetric rank also grows with the same exponent ω, so that ω can be computed in the symmetric setting, where it may be easier to determine.