Definition
Singular is an open‑source computer algebra system (CAS) specialized for polynomial computations, particularly in the fields of algebraic geometry, commutative algebra, and singularity theory. It provides algorithms for solving systems of polynomial equations, computing Gröbner bases, and analyzing local properties of algebraic varieties.
Overview
Developed initially in the early 1990s at the University of Kaiserslautern, Germany, Singular is distributed under the GNU General Public License. The program is written primarily in C and C++ and runs on major operating systems, including Linux, macOS, and Windows (via binary distributions or compilation from source). It is widely used in both academic research and education for tasks such as:
- Computing Gröbner bases and standard bases for ideals in polynomial rings.
- Performing primary decomposition of ideals.
- Analyzing singular points of algebraic varieties and determining multiplicities.
- Working with modules over polynomial rings, including syzygy computations.
- Interfacing with other mathematical software (e.g., GAP, SageMath) through libraries and exchange formats.
Singular’s user interface consists of an interactive command‑line interpreter, and it also offers a library API that allows integration into external programs. Documentation includes a comprehensive manual, numerous tutorial examples, and an extensive collection of research‑level algorithms.
Etymology / Origin
The name “Singular” reflects the system’s original focus on the study of singularities—points at which an algebraic variety fails to be smooth. The term “singular” in mathematics denotes such irregular points, and the software was conceived to provide computational tools for their investigation.
Characteristics
| Feature | Description |
|---|---|
| Core Language | A domain‑specific language designed for algebraic operations; supports loops, conditional statements, and user‑defined procedures. |
| Data Types | Polynomials, power series, matrices, modules, and ideals over various coefficient fields (e.g., rational numbers, finite fields, algebraic extensions). |
| Algorithms | Gröbner basis computation (Buchberger, F4, F5), standard basis for local orderings, primary decomposition, resolution of singularities, factorization, and integer matrix normal forms. |
| Extensibility | Plug‑in architecture; users can add new algorithms as external libraries written in C/C++. |
| Interoperability | Input/output in standard formats such as OpenMath, XML, and interfaces to SageMath, GAP, and CoCoA. |
| Performance | Optimized for large polynomial systems; recent versions incorporate parallelism for certain Gröbner‑basis calculations. |
| License | GNU GPL v2 or later, permitting free redistribution and modification. |
Related Topics
- Computer Algebra Systems – Other CAS platforms such as Mathematica, Maple, Magma, and SageMath.
- Gröbner Bases – A fundamental algorithmic tool for solving polynomial systems, central to Singular’s functionality.
- Algebraic Geometry – The mathematical discipline studying solutions to polynomial equations; Singular provides computational support for its methods.
- Commutative Algebra – The branch of algebra dealing with ideals and modules over rings; many of Singular’s algorithms stem from this area.
- Singularity Theory – The study of points where mathematical objects are not well‑behaved; Singular was originally created to aid research in this field.
- Open‑Source Mathematics Software – Initiatives promoting freely available mathematical tools, of which Singular is a prominent example.