documentation:latest:ideal:groebner

Available versions of this document: latest release, release 4.0, release 3.6, release 3.5, nightly master

Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

# BigObject Groebner

from application ideal

The Groebner object collects information about an ideal depending on the monomial ordering of the ambient ring.

Properties of an ideal with a monomial ordering from commutative algebra.

BASIS

The elements of the Groebner basis corresponding to the given order. This may vary for different algorithms, even if the order stays the same.

Type:
Array<Polynomial<Rational,Int>>
depends on extension:

INITIAL_FORMS

The initial forms of all polynomials in the BASIS, with respect to either the ORDER_VECTOR or the first row of the ORDER_MATRIX.

Type:
Array<Polynomial<Rational,Int>>
depends on extension:

INITIAL_IDEAL

The initial order corresponding to the given order. This is always a MONOMIAL ideal, even if only a weight vector is provided. Internally this weight vector will be concatenated with a total order.

Type:
Ideal
depends on extension:

Properties defining the Groebner subobject, i.e. a monomial order.

ORDER_MATRIX

The matrix defining the monomial ordering. For performance reasons this is realized via several weight vectors preceding a lexicographic order. (Singular: a(row), a(row),…,lp) Note that only one of ORDER_MATRIX, ORDER_VECTOR, ORDER_NAME should be given.

Type:
Matrix<Int,NonSymmetric>

ORDER_NAME

A string containing the name of the monomial ordering. Currently we follow the singular conventions, i.e. dp, lp, rp, ds, etc. Note that only one of ORDER_MATRIX, ORDER_VECTOR, ORDER_NAME should be given.

Type:
String

ORDER_VECTOR

A weight vector for the monomial ordering, a reverse lexicographic order will be used as tie-breaker. (Singular: wp(vector)) This vector is expected to consist of positive integers only. Note that only one of ORDER_MATRIX, ORDER_VECTOR, ORDER_NAME should be given.

Type:
Vector<Int>

Functions, methods and objects and attached from/to Singular.

SINGULAR_IDEAL

Intermediate object wrapping the Singular objects, i.e. the ring with the monomial ordering and the ideal.

Type:
SingularIdeal
from extension:

division

UNDOCUMENTED

from extension:

reduce(Polynomial p)

Reduce a Polynomial p with respect to the Groebner basis.

Parameters:

Polynomial p

Returns:
Polynomial
from extension:
reduce(Ideal I)

Reduce an Ideal I with respect to the Groebner basis.

Parameters:

Ideal I

Returns:
Array<Polynomial>
from extension:

• documentation/latest/ideal/groebner.txt