On an effective determination of a Shintani's decomposition of the cone \({\mathbb{R}}_ +^ n\) (Q1326738)

Let \(K\) be a totally real algebraic number field of degree \(n \geq 2\). The author develops a method for determining a fundamental domain for the action of the group \(E^ +_ k\) of totally positive units of \(K\) on \((\mathbb{R}^{ > 0})^ n\). The method can be viewed as a modification of Buchmann's generalization of Lagrange's algorithm, and it yields generators for \(E^ +_ k\) as a by-product. The computations are carried out in the maximal real subfield of the field of 11-th roots of unity as an example.