**Preface** (ps-file) |

** 0 Introductory Remarks on Computer Algebra** |

** 1 Basic Notations and Ideas: A Historical Account** |

** 2 Basic Computational Problems and Their Solution** |

2.1 The Geometry-Algebra Dictionary |

2.2 Basic Applications of Gröbner Bases |

** 3. An Introduction to Singular** |

3.1 General Remarks on Singular and its Syntax |

3.2 Rings in Singular |

3.2.1 Global Monomial Orders |

3.2.2 Creating Ring Maps |

3.3 Ideals, Vectors and Modules in Singular |

3.4 Handling Graded Modules |

3.5 Computing Gröbner Bases |

3.6 Basic Applications of Gröbner Bases (revisited) |

3.6.1 Ideal Membership Test |

3.6.2 Elimination |

3.6.3 Kernel of a Ring Map |

3.6.4 Test for Subalgebra Membership |

3.6.5 Test for Surjectivity of a Ring Map |

3.6.6 Syzygies and Free Resolutions |

3.7 Gröbner Bases over Noncommutative Algebras |

3.8 Writing Singular Procedures and Libraries |

3.9 Communication with Other Systems |

3.10 Visualization: Plotting Curves and Surfaces |

** Practical Session I** |

** Practical Session II** |

** 4 Homological Algebra I** |

4.1 Lifting Homomorphisms |

4.2 Constructive Module Theory |

4.2.1 Cokernels and Mapping Cones |

4.2.2 Modulo |

4.2.3 Kernel, Hom, Ext, Tor, and more |

4.2.4 Some Explicit Constructions |

** 5 Homological Algebra II** |

5.1 Flatness |

5.2 Depth and Codimension |

5.3 Cohen-Macaulay Rings |

** Practical Session III** |

** 6 Solving Systems of Polynomial Equations** |

6.1 Gröbner Basis Techniques |

6.1.1 Computing Dimension |

6.1.2 Zero-Dimensional Solving by Elimination |

6.1.3 Decomposition (Factorizing Buchberger Algorithm, Triangular Decompositions) |

6.2 Resultant Based Methods |

6.2.1 The Sylvester Resultant |

6.2.2 Multipolynomial Resultants |

6.2.3 Zero-Dimensional Solving via Resultants |

** 7 Primary Decomposition and Normalization** |

7.1 Primary Decomposition |

7.2 Normalization |

** Practical Session IV** |

** 8 Algorithms for Invariant Theory** |

8.1 Finite Groups |

8.1.1 The Nonmodular Case |

8.1.2 The Modular Case |

8.1.3 Quotients for Finite Group Actions |

8.2 Linearly Reductive Groups |

** 9 Computing in Local Rings** |

9.1 Rings Implemented by Monomial Orders |

9.2 Standard Bases and their Computation |

9.3 Factorization and Primary Decomposition |

9.4 Computing Dimension |

9.5 Elimination |

9.6 Hamburger-Noether Expansion |

** Practical Session V** |

** Appendix A. Sheaf Cohomology and Beilinson Monads** |

** Appendix B. Solutions to Exercises** |

** References** |

** Index** |