Some Methods for Evaluating the Regulator of a Real Quadratic Function Field

Abstract

We develop methods for the rapid computation of the regulator of a real quadratic congruence function field K = k(x)(√D). By extending Shanks' infrastructure ideas in real quadratic number fields to real quadratic congruence function fields we obtain a baby step–giant step method for evaluating the regulator of K in O(|D|¼) polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in O(|D|⅕) polynomial operations. By implementing both methods on a computer, we found that the O(|D|⅕) algorithm tends to be far better than the baby step–giant step algorithm in those cases where the regulator exceeds 10⁸.