The algebraic complexity of computing a family of bilinear forms

Some aspects of the theory of the algebraic complexity of computations are investigated, namely, the complexity of the computation of certain sets of bilinear forms the point of view of the number of multiplications and divisions. The complexity of the computation of a pair of bilinear forms is characterized. A new, close to linear, estimate is obtained for the complexity of computing a product of polynomials over a finite field. A group of non-singular linear tensor-rank-preserving transformations is described. The behavior almost everywhere of the rank in tensor space is considered.