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
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The Groebner object collects information about an ideal depending on the monomial ordering of the ambient ring.
Properties of Groebner
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Properties of an ideal with a monomial ordering from commutative algebra.
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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_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
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Properties defining the Groebner subobject, i.e. a monomial order.
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ORDER_MATRIX: common::Matrix<Int, NonSymmetric>
The matrix defining the monomial ordering.
Note that only one of ORDER_MATRIX, ORDER_VECTOR, ORDER_NAME should be given.
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ORDER_NAME: common::String
A string containing the name of the monomial ordering. Currently we follow the singular conventions, i.e. dp, lex, ds, etc.
Note that only one of ORDER_MATRIX, ORDER_VECTOR, ORDER_NAME should be given.
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ORDER_VECTOR: common::Vector<Int>
A weight vector for the monomial ordering.
Note that only one of ORDER_MATRIX, ORDER_VECTOR, ORDER_NAME should be given.
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Intermediate objects for interfacing with Singular.
Contained in extensionbundled:singular.-
SINGULAR_IDEAL: SingularIdeal
Intermediate object wrapping the Singular objects, i.e. the ring with the monomial ordering and the ideal.
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User Methods of Groebner
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An ideal in a polynomial ring.
Properties of Ideal
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Properties of an ideal computed via commutative algebra.
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DIM: common::Int
The dimension of the ideal, i.e. the Krull dimension of 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.
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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 -
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Properties defining an ideal.
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GENERATORS: common::Array<Polynomial<Rational, Int>>
A set of generators usually given by the user and not unique.
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User Methods of Ideal
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SOLVE ()
UNDOCUMENTED
Contained in extensionbundled:singular.
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User Functions
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functions attached from/to Singular.
Contained in extensionbundled:singular.-
singular_eval (s)
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singular_get_int (s)
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singular_get_var (s) → List
Property Types
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SingularIdeal
An intermediate object wrapping the ideal on the Singular side and providing its methods.
Contained in extensionbundled:singular.