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 $K=k(x)({\sqrt{D}})$. 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 $O( |D|\supfrac 14 )$ polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in $O( |D|\supfrac 15 )$ polynomial operations. By implementing both methods on a computer, we found that the $O( |D|\supfrac 15 )$-algorithm tends to be far better than the baby step-giant step algorithm in those cases where the regulator exceeds $10^8$.