Support
The SINGULAR Team provides free support.
Release
February 2010: Release of SINGULAR version 3-1-1. More
Jenks Prize
July 2004: The Richard D. Jenks Prize for Excellence in Software Engineering for Computer Algebra was awarded to the Singular team. More
Book: "A Singular Introduction to Commutative Algebra"
Table of contents:
| Preface (ps-file) |
| 1 Rings, Ideals and Standard Bases |
| 1.1 Rings, Polynomials and Ring Maps |
| 1.2 Monomial Orderings |
| 1.3 Ideals and Quotient Rings |
| 1.4 Local Rings and Localization |
| 1.5 Rings Associated to Monomial Orderings |
| 1.6 Normal Forms and Standard Bases |
| 1.7 The Standard Basis Algorithm |
| 1.8 Operations on Ideals and their Computation |
| 1.8.1 Ideal Membership |
| 1.8.2 Intersection with Subrings |
| 1.8.3 Zariski Closure of the Image |
| 1.8.4 Solvability of Polynomial Equations |
| 1.8.5 Solving Polynomial Equations |
| 1.8.6 Radical Membership |
| 1.8.7 Intersection of Ideals |
| 1.8.8 Quotient of Ideals |
| 1.8.9 Saturation |
| 1.8.10 Kernel of a Ring Map |
| 1.8.11 Algebraic Dependence and Subalgebra Membership |
| 2. Modules |
| 2.1 Modules, Submodules and Homomorphisms |
| 2.2 Graded Rings and Modules |
| 2.3 Standard Bases for Modules |
| 2.4 Exact Sequences and free Resolutions |
| 2.5 Computing Resolutions and the Syzygy Theorem |
| 2.6 Modules over Principal Ideal Domains |
| 2.7 Tensor Product |
| 2.8 Operations on Modules and their Computation |
| 2.8.1 Module Membership Problem |
| 2.8.2 Intersection with Free Submodules |
| 2.8.3 Intersection of Submodules |
| 2.8.4 Quotients of Submodules |
| 2.8.5 Radical and Zerodivisors of Modules |
| 2.8.6 Annihilator and Support |
| 2.8.7 Kernel of a Module Homomorphism |
| 2.8.8 Solving Systems of Linear Equations |
| 3. Noether Normalization and Applications |
| 3.1 Finite and Integral Extensions |
| 3.2 The Integral Closure |
| 3.3 Dimension |
| 3.4 Noether Normalization |
| 3.5 Applications |
| 3.6 An Algorithm to Compute the Normalization |
| 3.7 Procedures |
| 4. Primary Decomposition and Related Topics |
| 4.1 The Theory of Primary Decomposition |
| 4.2 Zero-dimensional Primary Decomposition |
| 4.3 Higher Dimensional Primary Decomposition |
| 4.4 The Equidimensional Part of an Ideal |
| 4.5 The Radical |
| 4.6 Procedures |
| 5. Hilbert Function and Dimension |
| 5.1 The Hilbert Function and the Hilbert Polynomial |
| 5.2 Computation of the Hilbert-Poincare Series |
| 5.3 Properties of the Hilbert Polynomial |
| 5.4 Filtrations and the Lemma of Artin-Rees |
| 5.5 The Hilbert-Samuel Function |
| 5.6 Characterization of the Dimension of Local Rings |
| 5.7 Singular Locus |
| 6. Complete Local Rings |
| 6.1 Formal Power Series Rings |
| 6.2 Weierstrass Preparation Theorem |
| 6.3 Completions |
| 6.4 Standard bases |
| 7. Homological Algebra |
| 7.1 Tor and Exactness |
| 7.2 Fitting Ideals |
| 7.3 Flatness |
| 7.4 Local Criteria for Flatness |
| 7.5 Flatness and Standard Bases |
| 7.6 Koszul Complex and Depth |
| 7.7 Cohen-Macaulay Rings |
| 7.8 Further Characterization of Cohen-Macaulayness |
| A. Geometric Background |
| A.1 Introduction by Pictures (ps-file) |
| A.2 Affine Algebraic Varieties |
| A.3 Spectrum and Affine Schemes |
| A.4 Projective Varieties |
| A.5 Projective Schemes and Varieties |
| A.6 Morphisms between Varieties |
| A.7 Projective Morphisms and Elimination |
| A.8 Local versus Global Properties |
| A.9 Singularities |
| B. SINGULAR - A Short Introduction (ps-file) |
| B.1 Downloading Instructions |
| B.2 Getting Started |
| B.3 Procedures and Libraries |
| B.4 Data Types |
| B.5 Functions |
| B.6 Control Structures |
| B.7 System Variables |
| B.8 Libraries |
| B.9 SINGULAR and Maple |
| B.10 SINGULAR and Mathematica |
| B.11 SINGULAR and MuPAD |
| References (ps-file) |
| Index (ps-file) |
| Algorithms |
| SINGULAR Examples |