Some properties of disjoint sums of tensors related to matrix multiplication

Let t be a disjoint sum of tensors associated to a matrix product. The rank of the s-th tensorial power of t can be bounded by an expression involving the elemnts of t exponent for matrix multiplicatio. This relation leads to a trascendental equation defining a new exponent for matrix multiplication. The use of this approach allowed reducing to 2.5166 the exponent 2.5218 due to V.Pan, S.Winograd [7,8] and A.Schonhage [9].