application: ideal
This application allows to define ideals and enables other applications to use these. For example we can compute the tropical variety of an ideal via gfan in the application 'tropical'. Using this application with the bundled extension Singular adds a lot more commutative algebra power.
Objects

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

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

BASIS: common::Array<Polynomial<Rational, Int>>
The elements of the Groebner basis corresponding to the given order. This may vary for different algorithms, even if the order stays the same.
Depends on: singular 
INITIAL_FORMS: common::Array<Polynomial<Rational, Int>>
The initial forms of all polynomials in the BASIS, with respect to either the ORDER_VECTOR or the first row of the ORDER_MATRIX.
Depends on: singular 
INITIAL_IDEAL: 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.
Depends on: singular


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

ORDER_MATRIX: common::Matrix<Int, NonSymmetric>
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.

ORDER_NAME: common::String
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.

ORDER_VECTOR: common::Vector<Int>
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 of ORDER_MATRIX, ORDER_VECTOR, ORDER_NAME should be given.


Functions, methods and objects and attached from/to Singular.
Contained in extensionsingular
.
SINGULAR_IDEAL: SingularIdeal
Intermediate object wrapping the Singular objects, i.e. the ring with the monomial ordering and the ideal.

User Methods of Groebner

division ()
UNDOCUMENTED
Contained in extensionsingular
. 
reduce (p) → Polynomial
Reduce a Polynomial p with respect to the Groebner basis.
Contained in extensionsingular
.


An ideal in a polynomial ring.
Properties of Ideal

Properties of an ideal computed via commutative algebra.


DIM: common::Int
The dimension of the ideal, i.e. the Krull dimension of Polynomial ring/Ideal.
Depends on: singular 
HILBERT_POLYNOMIAL: common::Polynomial<Rational, Int>
The Hilbert polynomial of the ideal. For toric ideals this is linked with the Ehrhart polynomial.




PRIMARY_DECOMPOSITION: common::Array<Ideal>
An array containing the primary decomposition of the given ideal, i.e. the contained ideals are PRIMARY and their intersection is the given ideal.
Depends on: singular 



Properties defining an ideal.

GENERATORS: common::Array<Polynomial<Rational, Int>>
A set of generators usually given by the user and not unique.

User Methods of Ideal

contains_monomial (s) → Polynomial
Check via saturation whether the ideal contains a monomial. Returns a monomial from the ideal or the trivial monomial if there is none.
Contained in extensionsingular
.Parameters
String s Optional term order (see ORDER_NAME) for intermediate Groebner bases, default: "dp"Returns
Polynomial 
SOLVE ()
UNDOCUMENTED
Contained in extensionsingular
.

User Functions

With these clients you can create ideals belonging to various parameterized families which occur frequently in comumutative algebra.

pluecker_ideal (d, n) → Ideal


Functions, methods and objects and attached from/to Singular.
Contained in extensionsingular
.
singular_eval (s)

singular_get_int (s)

singular_get_var (s) → List
Property Types

Functions, methods and objects and attached from/to Singular.
Contained in extensionsingular
.
SingularIdeal
An intermediate object wrapping the ideal on the Singular side and providing its methods.
