Some Computational Techniques

In the previous chapters, various methods for computing the regulator were presented. Using Shanks’ infrastructure, we have seen in Chapter 7 how to improve the continued fraction algorithm for computing the regulator using the baby-step giant-step method, resulting in an algorithm that computes R_Δ unconditionally in time O(Δ^1/4+∈) and, using improvements due to Buchmann, Williams, and Vollmer, in time O(R ^1/2_Δ Δ^∈). In Chapter 8 we have seen how the class number and regulator are intimately connected to L(1, χ) via the analytic class number formula (Corollary 8.35.1), and in Chapter 9 methods for efficiently computing estimates of h_Δ or h_ΔR_Δ using the analytic class number formula were discussed.