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Algorithms which are implemented in the SINGULAR kernel include the following:
- General standard basis algorithm: for any monomial ordering. Includes:
- Buchberger algorithm (well-orderings)
- Mora algorithm (tangent-cone orderings)
- Variants of standard basis algorithm:
- Traverso's Hilbert-driven Gröbner basis algorithm
- weighted-ecart-method
- high-corner-method
- factorizing Buchberger algorithm
- FGLM for change of monomial ordering
- Ideal Theory:
- Intersection, ideal quotient, elimination and saturation
- Syzygies and free resolutions of modules:
- Standard-basis driven, Schreyer's, and La Scala's algorithm
- Combinatorics:
- dimensions, Hilbert series, multiplicities, etc.
- Multivariate polynomial computations:
- gcd, resultant, factorization
- Characteristic sets:
- Wu's method