## 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 basis 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.
`fres`
  fres
  improved version of sres, computes a free resolution of an ideal or module using Schreyer's method. The input has to be a standard basis.
`lres`
  lres
  computes a free resolution of an ideal or module with LaScala'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.

  This is implemented as extended Euclidean Algorithm, and 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.

  Factorization of polynomials over algebraic extensions is provided by factoring the norm for univariate polynomials f (the gcd of f and the factors of the norm is a factorization of f) resp. by the extended Zassenhaus algorithm for multivariate polynomials.

`gcd`
  gcd
  computes greatest common divisors of univariate and multivariate polynomials.

  In the univariate case NTL is used. For prime characteristic, a subresultant gcd is used. In characteristic 0, the EZGCD is used, except for a special case where a modular algorithm is used.

`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
  finds all roots of a 0-dimensional ideal with multivariate resultants

### Controlling computations

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

User manual for Singular version 4.4.1, 2025, generated by texi2html.