3.4 Implemented algorithms

The basic algorithm in SINGULAR is a general standard basis algorithm for any monomial ordering which is compatible with the natural semi-group structure of the exponents. This includes well-orderings (Buchberger algorithm to compute a Groebner basis) and tangent cone orderings (Mora algorithm) as special cases.

Nonetheless, there are a lot of other important algorithms:

• Algorithms to compute the standard operations on ideals and modules: intersection, ideal quotient, elimination, etc.
• Different Syzygy algorithms and algorithms to compute free resolutions of modules.
• Combinatorial algorithms to compute dimensions, Hilbert series, multiplicities, etc.
• Algorithms for univariate and multivariate polynomial factorization, resultant and gcd computations.

Commands to compute standard bases

facstd

facstd
computes a list of Groebner bases via the Factorizing Groebner Basis Algorithm, i.e., their intersection has the same radical as the original ideal. It need not be a Groebner basis of the given ideal.

The intersection of the zero-sets is the zero-set of the given ideal.

fglm

fglm
computes a Groebner basis provided that a reduced Groebner basis w.r.t. another ordering is given.

Implements the so-called FGLM (Faugere, Gianni, Lazard, Mora) algorithm. The given ideal must be zero-dimensional.

groebner

groebner
computes a standard resp. Groebner bases using a heuristically chosen method.

This is the preferred method to compute a standard resp. Groebner bases.

mstd

mstd
computes a standard basis and a minimal set of generators.

std

std
computes a standard resp. Groebner basis.

stdfglm

stdfglm
computes a Groebner basis in a ring with a “difficult” ordering (e.g., lexicographical) via std w.r.t. a “simple” ordering and fglm.

The given ideal must be zero-dimensional.

stdhilb

stdhilb
computes a Groebner basis in a ring with a “difficult” ordering (e.g., lexicographical) via std w.r.t. a “simple” ordering and a std computation guided by the Hilbert series.

Further processing of standard bases

The next commands require the input to be a standard basis.

degree

degree
computes the (Krull) dimension, codimension and the multiplicity.

The result is only displayed on the screen.

dim

dim
computes the dimension of the ideal resp. module.

highcorner

highcorner
computes the smallest monomial not contained in the ideal resp. module. The ideal resp. module has to be finite dimensional as a vector space over the ground field.

hilb

hilb
computes the first, and resp. or, second Hilbert series of an ideal resp. module.

kbase

kbase
computes a vector space basis (consisting of monomials) of the quotient of a ring by an ideal resp. of a free module by a submodule.

The ideal resp. module has to be finite dimensional as a vector space over the ground field and has to be represented by a standard basis w.r.t. the ring ordering.

mult

mult
computes the degree of the monomial ideal resp. module generated by the leading monomials of the input.

reduce

reduce
reduces a polynomial, vector, ideal or module to its normal form with respect to an ideal or module represented by a standard basis.

vdim

vdim
computes the vector space dimension of a ring (resp. free module) modulo an ideal (resp. module).

Commands to compute resolutions

res

res
computes a free resolution of an ideal or module using a heuristically chosen method. This is the preferred method to compute free resolutions of ideals or modules.

lres

lres
computes a free resolution of an ideal or module with La Scala’s method. The input needs to be homogeneous.

mres

mres
computes a minimal free resolution of an ideal or module with the Syzygy method.

sres

sres
computes a free resolution of an ideal or module with Schreyer’s method. The input has to be a standard basis.

nres

nres
computes a free resolution of an ideal or module with the standard basis method.

syz

syz
computes the first Syzygy (i.e., the module of relations of the given generators).

Further processing of resolutions

betti

betti
computes the graded Betti numbers of a module from a free resolution.

minres

minres
minimizes a free resolution of an ideal or module.

regularity

regularity
computes the regularity of a homogeneous ideal resp. module from a given minimal free resolution.

Processing of polynomials

char_series

char_series
computes characteristic sets of polynomial ideals.

extgcd

extgcd
computes the extended gcd of two polynomials.

Implemented as extended Euclidean Algorithm. Applicable for univariate polynomials only.

factorize

factorize
computes factorization of univariate and multivariate polynomials into irreducible factors.

The most basic algorithm is univariate factorization in prime characteristic. The Cantor-Zassenhaus Algorithm is used in this case. For characteristic 0, a univariate Hensel-lifting is done to lift from prime characteristic to characteristic 0. For multivariate factorization in any characteristic, the problem is reduced to the univariate case first, then a multivariate Hensel-lifting is used to lift the univariate factorization.

Note that there is no factorization of polynomials over algebraic extensions of Q.

gcd

gcd
computes greatest common divisors of univariate and multivariate polynomials.

For prime characteristic, a subresultant gcd is used. In characteristic 0, a modular algorithm is used for the univariate case. For the multivariate case, the EZGCD is used.

Note that there is no gcd calculation for polynomials over algebraic extensions of Q.

resultant

resultant
computes the resultant of two univariate polynomials using the subresultant algorithm.

Multivariate polynomials are considered as univariate polynomials in the main variable (which has to be specified by the user).

vandermonde

vandermonde
interpolates a polynomial from its values at several points

Matrix computations

bareiss

bareiss
implements sparse Gauss-Bareiss method for elimination (matrix triangularization) in arbitrary integral domains.

det

det
computes the determinant of a square matrix.

For matrices with integer entries a modular algorithm is used. For other domains the Gauss-Bareiss method is used.

minor

minor
computes all minors (=subdeterminants) of a given size for a matrix.

Numeric computations

laguerre

laguerre
computes all (complex) roots of a univariate polynomial

uressolve

uressolve
find all roots of 0-dimensional ideal i with multivariate resultants

Controlling computations

option

option
allows setting of options for manipulating the behavior of computations (such as reduction strategies) and for showing protocol information indicating the progress of a computation.