BigObject Groebner [polymake wiki]

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====== BigObject Groebner ======
//from application [[..:ideal|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.
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{{anchor:basis:}}
  ?  **''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:
    :''[[..:common#Array |Array]]<[[..:common#Polynomial |Polynomial]]<[[..:common#Rational |Rational]],[[..:common#Int |Int]]%%>>%%''
    ? depends on extension:
    : [[:external_software|singular]]


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{{anchor:initial_forms:}}
  ?  **''INITIAL_FORMS''**
  :: The initial forms of all polynomials in the ''[[..:ideal:Groebner#BASIS |BASIS]]'', with respect to either the ''[[..:ideal:Groebner#ORDER_VECTOR |ORDER_VECTOR]]'' or the first row of the ''[[..:ideal:Groebner#ORDER_MATRIX |ORDER_MATRIX]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Polynomial |Polynomial]]<[[..:common#Rational |Rational]],[[..:common#Int |Int]]%%>>%%''
    ? depends on extension:
    : [[:external_software|singular]]


----
{{anchor:initial_ideal:}}
  ?  **''INITIAL_IDEAL''**
  :: The initial order corresponding to the given order. This is always a [[..:ideal:Ideal#MONOMIAL |MONOMIAL]] ideal, even if only a weight vector is provided. Internally this weight vector will be concatenated with a total order.
    ? Type:
    :''[[..:ideal:Ideal |Ideal]]''
    ? depends on extension:
    : [[:external_software|singular]]


----

==== Input properties ====
 Properties defining the Groebner subobject, i.e. a monomial order.
----
{{anchor:order_matrix:}}
  ?  **''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 ''[[..:ideal:Groebner#ORDER_MATRIX |ORDER_MATRIX]]'', ''[[..:ideal:Groebner#ORDER_VECTOR |ORDER_VECTOR]]'', ''[[..:ideal:Groebner#ORDER_NAME |ORDER_NAME]]'' should be given.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:order_name:}}
  ?  **''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 ''[[..:ideal:Groebner#ORDER_MATRIX |ORDER_MATRIX]]'', ''[[..:ideal:Groebner#ORDER_VECTOR |ORDER_VECTOR]]'', ''[[..:ideal:Groebner#ORDER_NAME |ORDER_NAME]]'' should be given.
    ? Type:
    :''[[..:common#String |String]]''


----
{{anchor:order_vector:}}
  ?  **''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 ''[[..:ideal:Groebner#ORDER_MATRIX |ORDER_MATRIX]]'', ''[[..:ideal:Groebner#ORDER_VECTOR |ORDER_VECTOR]]'', ''[[..:ideal:Groebner#ORDER_NAME |ORDER_NAME]]'' should be given.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Int |Int]]>''


----

==== Singular interface ====
 Functions, methods and objects and attached from/to Singular.
----
{{anchor:singular_ideal:}}
  ?  **''SINGULAR_IDEAL''**
  :: Intermediate object wrapping the Singular objects, i.e. the ring with the monomial ordering and the ideal.
    ? Type:
    :''[[..:ideal#SingularIdeal |SingularIdeal]]''
    ? from extension:
    : [[:external_software|bundled:singular]]


----
===== Methods =====

==== no category ====
{{anchor:division:}}
  ?  **''division''**
  ::UNDOCUMENTED
    ? from extension:
    : [[:external_software|bundled:singular]]


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{{anchor:reduce:}}
  ?  **''reduce([[..:common#Polynomial |Polynomial]] p)''**
  :: Reduce a ''[[..:common#Polynomial |Polynomial]]'' //p// with respect to the Groebner basis.
    ? Parameters:
    :: ''[[..:common#Polynomial |Polynomial]]'' ''p''
    ? Returns:
    :''[[..:common#Polynomial |Polynomial]]''
    ? from extension:
    : [[:external_software|bundled:singular]]
  ?  **''reduce([[..:ideal:Ideal |Ideal]] I)''**
  :: Reduce an ''[[..:ideal:Ideal |Ideal]]'' //I// with respect to the Groebner basis.
    ? Parameters:
    :: ''[[..:ideal:Ideal |Ideal]]'' ''I''
    ? Returns:
    :''[[..:common#Array |Array]]<[[..:common#Polynomial |Polynomial]]>''
    ? from extension:
    : [[:external_software|bundled:singular]]


----