Algorithmic Algebra and Number Theory

1. Front Matter

   Pages I-VIII

   PDF

2. Algorithmic Algebraic Number Theory

   1. Front Matter

      Pages 1-1

      PDF

   2. Sieving Methods for Class Group Computation

      • Johannes Buchmann, Michael J. Jacobson Jr., Stefan Neis, Patrick Theobald, Damian Weber

      Pages 3-10

   3. Arithmetic of Modular Curves and Applications

      • Gerhard Frey, Michael Müller

      Pages 11-48

   4. Local and Global Ramification Properties of Elliptic Curves

      • Ernst-Ulrich Gekeler

      Pages 49-64

   5. Techniques for the Computation of Galois Groups

      • Alexander Hulpke

      Pages 65-77

   6. Fortschritte in der inversen Galoistheorie

      • B. Heinrich Matzat

      Pages 79-102

   7. From Class Groups to Class Fields

      • Michael E. Pohst

      Pages 103-119

   8. A Gross-Zagier Formula for Function Fields

      • Hans-Georg Rück, Ulrich Tipp

      Pages 121-137

   9. Extremal Lattices

      • Rudolf Scharlau, Rainer Schulze-Pillot

      Pages 139-170

3. Algorithmic Commutative Algebra and Algebraic Geometry

   1. Front Matter

      Pages 171-171

      PDF

   2. On the Real Nullstellensatz

      • Eberhard Becker, Joachim Schmid

      Pages 173-185

   3. Primary Decomposition: Algorithms and Comparisons

      • Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister

      Pages 187-220

   4. Real Quantifier Elimination in Practice

      • Andreas Dolzmann, Thomas Sturm, Volker Weispfenning

      Pages 221-247

   5. Hilbert Series and Degree Bounds in Invariant Theory

      • Gregor Kemper

      Pages 249-263

   6. Invariant Rings and Fields of Finite Groups

      • Gregor Kemper, Gunter Malle

      Pages 265-281

   7. Computing Versal Deformations with Singular

      • Bernd Martin

      Pages 283-293

   8. Algorithms for the Computation of Free Resolutions

      • Thomas Siebert

      Pages 295-310

4. Algorithmic Group and Representation Theory

   1. Front Matter

      Pages 311-311

      PDF

   2. Computational Aspects of the Isomorphism Problem

      • Frauke M. Bleher, Wolfgang Kimmerle, Klaus W. Roggenkamp, Martin Wursthor

      Pages 313-329

This book contains 22 lectures presented at the final conference of the Ger­ man research program (Schwerpunktprogramm) Algorithmic Number The­ ory and Algebra 1991-1997, sponsored by the Deutsche Forschungsgemein­ schaft. The purpose of this research program and of the meeting was to bring together developers of computer algebra software and researchers using com­ putational methods to gain insight into experimental problems and theoret­ ical questions in algebra and number theory. The book gives an overview on algorithmic methods and on results ob­ tained during this period. This includes survey articles on the main research projects within the program: • algorithmic number theory emphasizing class field theory, constructive Galois theory, computational aspects of modular forms and of Drinfeld modules • computational algebraic geometry including real quantifier elimination and real algebraic geometry, and invariant theory of finite groups • computational aspects of presentations and representations of groups, especially finite groups of Lie type and their Heeke algebras, and of the isomorphism problem in group theory. Some of the articles illustrate the current state of computer algebra sys­ tems and program packages developed with support by the research pro­ gram, such as KANT and LiDIA for algebraic number theory, SINGULAR, RED LOG and INVAR for commutative algebra and invariant theory respec­ tively, and GAP, SYSYPHOS and CHEVIE for group theory and representation theory.

• algebra
• algebraic geometry
• algorithms
• computation
• computer algebra
• geometry
• linear optimization
• number theory

• Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, der Universität Heidelberg, Heidelberg, Germany

  B. Heinrich Matzat

• Fachbereich Mathematik, Universität Kaiserslautern, Kaiserslautern, Germany

  Gert-Martin Greuel

• Lehrstuhl D für Mathematik, RWTH Aachen, Aachen, Germany

  Gerhard Hiss

Policies and ethics