Promotionen in der Computeralgebra

Abstract

Maximilian Boy: On the Second Class Group of Real Quadratic Number Fields Betreuer: Gunter Malle (Kaiserslautern) ¨ ¨ Zweitgutachter: Jurgen Kluners (Paderborn) Februar 2012 https://kluedo.ub.uni-kl.de/frontdoor/ index/index/docId/2885 Zusammenfassung: This thesis generalizes the Cohen-Lenstra heuristic for the class groups of real quadratic number fields to higher class groups. A 'good part' of the second class group is defined. In general this is a non abelian proper factor group of the second class group. Properties of those groups are described, a probability distribution on the set of those groups is introduced and proposed as generalization of the Cohen-Lenstra heuristic for real quadratic number fields. The calculation of number field tables which contain information about higher class groups is explained and the tables are compared to the heuristic. The agreement is close. A program which can create an internet database for number field tables is presented. Christian Eder: Signature-based algorithms to compute standard bases Betreuer: Gerhard Pfister (Kaiserslautern) Zweitgutachter: Vladimir Gerdt (Dubna) April 2012 http://www.mathematik.uni-kl.de/~ederc Zusammenfassung: Standard bases are one of the main tools in computational commutative algebra. In 1965 Buchberger presented a criterion for such bases and thus was able to introduce a first approach for their computation. Since the basic version of this algorithm is rather inefficient due to the fact that it processes lots of useless data during its execution, active research for improvements of those kind of algorithms is quite important. In this thesis we introduce the reader to the area of computational commutative algebra with a focus on so-called signature-based standard basis algorithms. We do not only present the basic version of Buchberger's algorithm, but give an extensive discussion of different attempts optimizing standard basis computations, from several sorting algorithms for internal data up to different reduction processes. Afterwards the reader gets a complete introduction to the origin of signature-based algorithms in general, explaining the underlying ideas in detail. Furthermore, we give an extensive discussion in terms of correctness, termination, and efficiency, presenting various different variants of signature-based standard basis algorithms. Whereas Buchberger and others found criteria to discard useless computations which are completely based on the polynomial structure of the elements considered, Faug` ere presented a first signature-based algorithm in 2002, the F5 Algorithm. This algorithm is famous for generating much less computational overhead during its execution. Within this thesis we not only present Faug` ere's ideas, we also generalize them and end up with several different, optimized variants of his criteria for detecting redundant data. Being not completely focussed on theory, we also present information about practical aspects, comparing the performance of various implementations of those algorithms in the computer algebra system S INGULAR over a wide range of example sets. In the end we give a rather extensive overview of recent research in this area of computational commutative algebra.