On the period length of the generalized Lagrange algorithm (Q1820815)

The author previously obtained (see the preceding review) a number geometric generalization of Lagrange's continued fraction algorithm. This new algorithm yields a fundamental system of units and the class number of any algebraic number field F by means of computing cycles of reduced ideals. In the paper under review it is shown that the cardinality of a cycle of reduced ideals in an ideal class of an order \({\mathfrak O}\) of F is O(R) where R is the regulator of \({\mathfrak O}\), and where the big O-constant depends only on the degree of F over \({\mathbb{Q}}\). A lower bound for this cardinality is also given.