Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form algorithm

We show that the integers in the HMM LLL HNF algorithm have bit length O(m.log(m.B)), where m is the number of rows and B is the maximum square length of a row of the input matrix. This is only a little worse than the estimate O(m.log(B)) in the LLL algorithm.Comment: 10 page

On the computation of the values of zeta functions of totally real cubic fields

AbstractBased on earlier papers of the first author we give a concise formula for the values of class zeta functions of totally real cubic fields at even positive integers which is the exact analogue of the Barn-Siegel formula for real quadratic fields. For this purpose we use a rather complicated series representation for the aforementioned values depending on a parameter x which is analyzed for x → 0. The final formula is well suited for actual computations; two tables of values of class zeta functions are given at the end of the paper

Computation of 2-groups of positive classes of exceptional number fields

We present an algorithm for computing the 2-group of the positive divisor classes of a number field F in case F has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel WK2(F) in K2(F) for such number fields

On solving norm equations in global function fields

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.The potential of solving norm equations is crucial for a variety of applications of algebraic number theory, especially in cryptography. In this article we develop general effective methods for that task in global function fields for the first time

Diophantine equations over global function fields I: The Thue equation

AbstractWe solve completely Thue equations in function fields over arbitrary finite fields. In the function field case such equations were formerly only solved over algebraically closed fields (of characteristic zero and positive characteristic). Our method can be applied to similar types of Diophantine equations, as well