Dirichlet Groups and Lattice Reduction

Abstract

The main reason for the periodicity of algebraic multidimensional sails, is a regular structure of the corresponding Dirichlet groups. The simplest case of two-dimensional Dirichlet groups was studied in Chap. 8 in the first part of this book. In this chapter we study the general multidimensional case. We start with formulation of Dirichlet’s unity theorem on the structure of the group of units in orders. Further we describe the relation between Dirichlet groups and groups of units. Then we show how to calculate bases of the Dirichlet groups and positive Dirichlet groups respectively. Finally we briefly discuss the LLL-algorithm on lattice reduction which helps to decrease the computational complexity in many problems related to lattices (including the calculation of Dirichlet group bases).