Objects

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  Groebner

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

  Properties of Groebner

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    Commutative algebra

    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.

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      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.

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    Input properties

    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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  Ideal

  An ideal in a polynomial ring.

  Properties of Ideal

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    Commutative algebra

    Properties of an ideal computed via commutative algebra.

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      DEPTH: common::Int
      

      The depth of the ideal.

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      DIM: common::Int
      

      The dimension of the ideal, i.e. the Krull dimension of RING/Ideal.

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      GROEBNER: Groebner
      

      Subobject containing properties that depend on the monomial ordering of the ring.

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      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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      HOMOGENEOUS: common::Bool
      

      True if the ideal can be generated by homogeneous polynomials.

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      MONOMIAL: common::Bool
      

      True if the ideal can be generated by monomials.

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      PRIMARY: common::Bool
      

      True if the ideal is a primary ideal. I.e. its RADICAL is PRIME and in the quotient ring by the ideal every zero divisor is nilpotent.

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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.

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      PRIME: common::Bool
      

      True if the is ideal a prime ideal.

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      RADICAL: Ideal
      

      The radical of the ideal.

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      RING: common::Ring<Rational, Int>
      

      The ambient (polynomial) ring containing the ideal.

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      ZERO: common::Bool
      

      True if the ideal is the zero ideal.

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    Input properties

    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.