====== BigObject Ideal ======
//from application [[..:ideal|ideal]]//\\
\\
 An ideal in a polynomial ring.

===== Properties =====

==== Commutative algebra ====
 Properties of an ideal computed via commutative algebra.
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{{anchor:depth:}}
  ?  **''DEPTH''**
  :: The depth of the ideal.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? depends on extension:
    : [[:external_software|singular]]


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{{anchor:dim:}}
  ?  **''DIM''**
  :: The dimension of the ideal, i.e. the Krull dimension of Polynomial ring/Ideal.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? depends on extension:
    : [[:external_software|singular]]


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{{anchor:groebner:}}
  ?  **''GROEBNER''**
  :: Subobject containing properties that depend on the monomial ordering of the ring.
    ? Type:
    :''[[..:ideal:Groebner |Groebner]]''
    ? depends on extension:
    : [[:external_software|singular]]


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{{anchor:hilbert_polynomial:}}
  ?  **''HILBERT_POLYNOMIAL''**
  :: The Hilbert polynomial of the ideal. For toric ideals this is linked with the Ehrhart polynomial.
    ? Type:
    :''[[..:common#Polynomial |Polynomial]]<[[..:common#Rational |Rational]],[[..:common#Int |Int]]>''


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{{anchor:homogeneous:}}
  ?  **''HOMOGENEOUS''**
  :: True if the ideal can be generated by homogeneous polynomials.
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:monomial:}}
  ?  **''MONOMIAL''**
  :: True if the ideal can be generated by monomials.
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:n_variables:}}
  ?  **''N_VARIABLES''**
  :: The number of variables of the polynomial ring containing the ideal.
    ? Type:
    :''[[..:common#Int |Int]]''


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{{anchor:primary:}}
  ?  **''PRIMARY''**
  :: True if the ideal is a primary ideal. I.e. its ''[[..:ideal:Ideal#RADICAL |RADICAL]]'' is ''[[..:ideal:Ideal#PRIME |PRIME]]'' and  in the quotient ring by the ideal every zero divisor is nilpotent.
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:primary_decomposition:}}
  ?  **''PRIMARY_DECOMPOSITION''**
  :: An array containing the primary decomposition of the given ideal, i.e. the contained ideals are ''[[..:ideal:Ideal#PRIMARY |PRIMARY]]'' and their intersection is the given ideal.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:ideal:Ideal |Ideal]]>''
    ? depends on extension:
    : [[:external_software|singular]]


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{{anchor:prime:}}
  ?  **''PRIME''**
  :: True if the is ideal a prime ideal.
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:radical:}}
  ?  **''RADICAL''**
  :: The radical of the ideal.
    ? Type:
    :''[[..:ideal:Ideal |Ideal]]''
    ? depends on extension:
    : [[:external_software|singular]]


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{{anchor:zero:}}
  ?  **''ZERO''**
  :: True if the ideal is the zero ideal.
    ? Type:
    :''[[..:common#Bool |Bool]]''


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==== Input properties ====
 Properties defining an ideal.
----
{{anchor:generators:}}
  ?  **''GENERATORS''**
  :: A set of generators usually given by the user and not unique.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Polynomial |Polynomial]]<[[..:common#Rational |Rational]],[[..:common#Int |Int]]%%>>%%''


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===== Methods =====

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


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{{anchor:solve:}}
  ?  **''SOLVE''**
  ::UNDOCUMENTED
    ? from extension:
    : [[:external_software|bundled:singular]]


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{{anchor:contains_monomial:}}
  ?  **''contains_monomial([[..:common#String |String]] s)''**
  :: Check via saturation whether the ideal contains a monomial. Returns a monomial from the ideal or the trivial monomial if there is none.
    ? Parameters:
    :: ''[[..:common#String |String]]'' ''s'': Optional term order (see ''[[..:ideal:Groebner#ORDER_NAME |ORDER_NAME]]'') for intermediate Groebner bases, default: "dp"
    ? Returns:
    :''[[..:common#Polynomial |Polynomial]]''
    ? from extension:
    : [[:external_software|bundled:singular]]


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