Differential Methods

Abstract

We will investigate two problems which will prove to be of particular importance for the rest of the book. The first question is how much divisions may help for the computation of a set of polynomials. The example of the univariate polynomial f = X ³¹ shows that divisions indeed can help. Strassen discovered in 1973 [498] a technique for transforming a straight-line program for a set of rational functions to a division free straight-line program for the “coefficients” of the Taylor series of these functions (Thm. (7.1)). In particular, he showed that divisions do not help for the computation of a set of quadratic forms. This result, which marks the beginning of the theory of bilinear complexity, will be exploited later in Chap. 14. Following Baur and Strassen [32] we will show in the second part of the present chapter how to transform a straight-line program for computing a multivariate rational function into one that computes this function and its gradient. Combined with other lower bound techniques, such as Strassen¹ s degree bound introduced in the next chapter, this so-called derivative inequality (7.7) allows us to derive sharp lower bounds for the nonscalar complexity of numerous computational problems. The results of this chapter will also be used extensively in Chap. 16 where we will show that several computational problems in linear algebra are about as hard as matrix multiplication.