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

  ? Specializations:
  :: ''Ideal::Binomial'':  A binomial ideal represents an ideal which is generated by polynomials of the form p(X) - q(X), where p(X) and q(X) are both multivariate monomials. For example, x1*x2^2 - x1x3x4^10 would be a polynomial of this form, but x1^2 + x1 and  2x1 - x3 are not polynomials of this form. Toric ideals of lattice polytopes are one example of an ideal which may be represented by such a generating set. Since these generator sets have a special form, they may be represented compactly with a matrix.
===== 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''**
  ::
    ? Type:
    :''[[..:ideal:Groebner |Groebner]]''
    ? depends on extension:
    : [[:external_software|singular]]
    ? Properties of GROEBNER:
    :
      ?  **''BINOMIAL_BASIS''**
      :: An integer matrix representation of a binomial groebner basis. Rows correspond to polynomials, and columns to variables. For example, the row (1, -3, -1, 0, 2) corresponds to the polynomial x0*x4^2 - x^2*x3. 
        ? Type:
        :''[[..:common#Matrix |Matrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{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]]''


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


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


----
{{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]]''


----
{{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]]


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


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


----
{{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]]%%>>%%''


----

==== no category ====
{{anchor:binomial:}}
  ?  **''BINOMIAL''**
  ::UNDOCUMENTED
    ? Type:
    :''[[..:common#Bool |Bool]]''


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{{anchor:binomial_generators:}}
  ?  **''BINOMIAL_GENERATORS''**
  :: An integer matrix representation of a generating set of the binomial ideal. Rows correspond to polynomials, and columns to variables. The absolute value of an entry determines the degree of the coefficient of the corresponding column variable in the row polynomial. The parity determines whether it is in the positive or negative monomial. For example, the row (1, -3, -1, 0, 2) corresponds to the polynomial x0*x4^2 - x^2*x3.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>''
    ? Example:
    :: The following declares a binomial ideal via its matrix encoding,          and reencodes it into polynomials.
    :: <code perl> > $mat = new Matrix<Int>([1,2,0,-4],[3,1,0,1],[-4,-3,0,0]);
 > $ideal = new Ideal(BINOMIAL_GENERATORS=>$mat);
</code>
    ::  print $ideal->GENERATORS; x_0*x_1^2 - x_3^4 x_0^3*x_1*x_3 - 1 - x_0^4*x_1^3 + 1


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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]]


----