SINGULAR
Summary
SINGULAR is a Computer Algebra system for polynomial computations in commutative algebra, algebraic geometry, and singularity theory. SINGULAR‘s main computational objects are ideals and modules over a large variety of baserings. The baserings are polynomial rings over a field (e.g., finite fields, the rationals, floats, algebraic extensions, transcendental extensions), or localizations thereof, or quotient rings with respect to an ideal. SINGULAR features fast and general implementations for computing Groebner and standard bases, including e.g. Buchberger’s algorithm and Mora’s Tangent Cone algorithm. Furthermore, it provides polynomial factorizations, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, and many more related functionalities. Based on an easy-to-use interactive shell and a C-like programming language, SINGULAR‘s internal functionality is augmented and user-extendible by libraries written in the SINGULAR programming language. A general and efficient implementation of communication links allows SINGULAR to make its functionality available to other programs.
Authors
Gert-Martin Greuel, Gerhard Pfister, Hans Schönemann
Vendor
Centre for Computer Algebra, University of Kaiserslautern
Status
officially approved by the authorsAims and scope
Mathematical Classification
- 02.04 Commutative rings and algebras
- 02.04.01 Ideals, modules, homomorphisms
- 02.04.02 Polynomial and power series rings
- 02.04.03 Special rings
- 02.04.04 Graded rings and Hilbert functions
- 02.04.05 Integral dependence and normalization
- 02.04.06 Dimension theory
- 02.04.07 Factorization and primary decomposition
- 02.04.08 Syzygies and resolutions
- 02.04.09 Differential algebra
- 02.04.10 Groebner bases
- 02.05 Linear and multilinear algebra; matrix theory
- 02.05.01 Linear equations
- 02.05.02 Eigenvalues, singular values, and eigenvectors
- 02.05.03 Canonical forms
- 02.05.05 Integral matrices
- 02.05.06 Multilinear algebra
- 02.07 Category theory; homological algebra
- 02.08 Group theory and generalizations
- 03.01 Algebraic geometry
- 03.01.01 Local theory, singularities
- 03.01.02 Cycles and subschemes
- 03.01.03 Families, fibrations
- 03.01.04 Birational theory
- 03.01.05 Co(homology) theory
- 03.01.06 Arithmetic problems
- 03.01.08 Algebraic groups and geometric invariant theory
- 03.01.09 Special varieties
- 03.01.10 Real algebraic geometry
- 03.03 Several complex variables and analytic spaces
- 03.10 Visualization
Keywords
- Alexander polynomials
- algebraic dependences
- algebraic geometric codes
- algebraic geometry
- annihilators
- Arnold action
- associated primes
- Bernstein polynomials
- betti numbers
- Bigatti-La Scala-Robbiano algorithm
- branches of isolated space curve singularities
- Brieskorn lattices
- Brill-Noether algorithm
- Buchberger algorithm
- characteristic sets
- classification of singularities
- coding theory
- Cohen-Macaulay module
- Cohen-Macaulay ring
- cohomologies
- commutative algebra
- commutative algebras
- conductors
- Conti-Traverso algorithm
- critical points
- cup products
- curve singularities
- cyclic roots
- deformation of singularities
- deformations
- delta invariants
- derivatives
- determinants
- Di Biase-Urbanke algorithm
- dimensions
- eigenvalues
- elimination
- emacs interface
- equisingularity ideals
- equisingular strata
- esingular
- evaluation of logical expressions
- ext groups
- finite fields
- fitting ideals
- flattening stratification
- fractal walk algorithm
- free modules
- free resolution of modules
- Galois groups
- Gauss algorithm
- Gauss-Manin connection
- gcd
- genus of an algebraic curve
- global orderings
- gmp
- greatest common divisors
- Groebner bases
- Groebner walk
- Hamburger-Noether development
- Hamburger-Noether expansion
- Hessenberg form
- Hilbert function
- Hilbert series
- hom
- homological algebra
- homology
- homomorphism
- Hosten-Sturmfels algorithm
- hypersurfaces
- ideal memberships
- ideal operations
- ideal quotients
- ideals
- integer programming
- interpolation
- intersection multiplicity
- invariant rings
- invariants of a finite group action
- invariant theory
- isolated complete intersection singularities
- Jacobian matrices
- Jordan bases
- Jordan matrices
- Jordan normal forms
- kernels
- Kodaira-Spencer maps
- Koszul complex
- Laguerre solve
- lcm
- least common multiples
- linear algebra
- linear codes
- local orderings
- long coefficients
- matrices
- matrix
- matrix orderings
- maximal ideals
- Milnor code
- Milnor numbers
- minimal decompositions
- minimal polynomials
- mixed hodge structure
- module homomorphisms
- module operations
- module orderings
- module ranks
- modules
- monodromy
- monomial orderings
- mp
- multiplicity sequences
- multivariate polynomial factorization
- Newton non-degenerate
- Newton polygons
- Newton polytopes
- Noether normalization
- non-commutative rings
- normal forms
- normalization
- normalizations
- opposite hodge filtration
- orbit varieties
- parametrization
- perturbation walk
- plural
- polar curves
- polynomial computations
- polynomial factorization
- polynomial rings
- polynomials
- primary decompositions
- product orderings
- puiseux expansions
- puiseux pairs
- radicals
- random number generators
- reduced standard bases
- Rees algebra
- Rees closure
- relative orbit varieties
- resolution of singularities
- resultants
- Reynolds image
- Reynolds operator
- rings
- saturation
- singular
- singularities
- singularity theory
- space curves
- sparse matrices
- sparse polynomials
- spectral pairs
- spectrum of singularities
- splitting lemma
- squarefree
- standard bases
- stratification
- surf
- sv-decoding algorithm
- symbolic-numerical solving
- syzygies
- tangent cones
- tensor products
- term orderings
- Tjurina numbers
- toric ideals
- tran algorithm
- triangular systems
- univariate polynomial factorization
- v-filtration
- visualization
- Weierstrass semigroups
- weight filtration