The lower bounds on the additive complexity of bilinear problems in terms of some algebraic quantities

Until very recently, the lower bounds on the additive complexity of intensively studied linear and bilinear arithmetic algorithms for arithmetic computational problems have relied on the active operation-basic substitution argument. Consequently, these bounds have not exceeded the dimension of the problems that is the total number of input variables and outputs. Another approach to the problem follows from the method presented by J. Morgenstern. A third approach reduces the problem to the study of a strong regularity of matrices. Mathematical notation.