Computing Shintani domains

Abstract

For a number field [Formula: see text] having at least one embedding into the real numbers we give an algorithm to obtain a Shintani domain for the action of the totally positive units of [Formula: see text] under its geometric embedding. Our algorithm modifies a known signed, or virtual, fundamental domain until it becomes a true one. We examine the results of extensive runs of our algorithm for cubic, quartic and quintic number fields. We also give some data for totally real sextic fields where our algorithm is still often practical.