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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
Commutative algebra
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:
 depends on extension:

INITIAL_FORMS
The initial forms of all polynomials in the
BASIS
, with respect to either theORDER_VECTOR
or the first row of theORDER_MATRIX
. Type:
 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:
 depends on extension:
Input properties
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:

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:

ORDER_VECTOR
A weight vector for the monomial ordering, a reverse lexicographic order will be used as tiebreaker. (Singular:
wp(vector)
) This vector is expected to consist of positive integers only. Note that only one ofORDER_MATRIX
,ORDER_VECTOR
,ORDER_NAME
should be given. Type:
Singular interface
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:
 from extension:
Methods
no category

division
UNDOCUMENTED
 from extension:

reduce(Polynomial p)
Reduce a
Polynomial
p with respect to the Groebner basis. Parameters:
 Returns:
 from extension:

reduce(Ideal I)
Reduce an
Ideal
I with respect to the Groebner basis. Parameters:
Ideal
I
 Returns:
 from extension: