Certain systems of bilinear forms whose minimal algorithms are all quadratic

In this paper we prove that every minimal algorithm for computing a bilinear form is quadratic. It is then easy to show that for certain systems of bilinear forms all minimal algorithms are quadratic. One such system is for computing the product of two arbitrary elements in a finite algebraic extension field. This result, together with the results of the author in (E. Feig, to appear) and those of Winograd (Theoret. Comput. Sci.8 (1979), 359–377), completely characterize all minimal algorithms for computing products in these fields.