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Algorithms in the SINGULAR Kernel
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
One of the main design goals of SINGULAR is efficiency. And, indeed, SINGULAR is one of the fastest CAS for polynomial computations.

KL, 06/03 http://www.singular.uni-kl.de