Multiplicative and Bilinear Complexity

Abstract

This chapter introduces the framework of the theory of bilinear complexity and is the basis for the study of Chap. 15–20. The language and concepts introduced here will, e. g., allow for a concise treatment of the fast matrix multiplication algorithms which we will discuss in the next chapter. We shall first turn our attention to the computation of a set of quadratic polynomials. According to Cor. (7.5) divisions do not help for the computation of such polynomials (at least when the field is infinite). The proof of that theorem in the case of quadratic polynomials implies a different characterization of the multiplicative complexity of these polynomials. We then associate to a set of quadratic polynomials quadratic maps between finite dimensional k-spaces and study the maps rather than the polynomials; this gives rise to the notion of computation for such maps. Specializing the quadratic maps further to bilinear maps and modifying the computations will lead to the important notion of rank or bilinear complexity of a bilinear map.