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 |