Class Ideal<C extends GcdRingElem<C>>

java.lang.Object
edu.jas.application.Ideal<C>
All Implemented Interfaces:
Serializable, Comparable<Ideal<C>>
public class Ideal<C extends GcdRingElem<C>> extends Object implements Comparable<Ideal<C>>, Serializable
Ideal implements some methods for ideal arithmetic, for example intersection, quotient and zero and positive dimensional ideal decomposition.
See Also:

  • Field Details

    • logger

      private static final org.apache.logging.log4j.Logger logger

    • debug

      private static final boolean debug

    • list

      protected PolynomialList<C extends GcdRingElem<C>> list
      The data structure is a PolynomialList.

    • isGB

      protected boolean isGB
      Indicator if list is a Groebner Base.

    • testGB

      protected boolean testGB
      Indicator if test has been performed if this is a Groebner Base.

    • isTopt

      protected boolean isTopt
      Indicator if list has optimized term order.

    • bb

      protected final GroebnerBaseAbstract<C extends GcdRingElem<C>> bb
      Groebner base engine.

    • red

      protected final Reduction<C extends GcdRingElem<C>> red
      Reduction engine.

    • engine

      protected final SquarefreeAbstract<C extends GcdRingElem<C>> engine
      Squarefree decomposition engine.

  • Constructor Details

    • Ideal

      public Ideal(GenPolynomialRing<C> ring)
      Constructor.
      Parameters:
      ring - polynomial ring

    • Ideal

      public Ideal(GenPolynomialRing<C> ring, List<GenPolynomial<C>> F)
      Constructor.
      Parameters:
      ring - polynomial ring
      F - list of polynomials

    • Ideal

      public Ideal(GenPolynomialRing<C> ring, List<GenPolynomial<C>> F, boolean gb)
      Constructor.
      Parameters:
      ring - polynomial ring
      F - list of polynomials
      gb - true if F is known to be a Groebner Base, else false

    • Ideal

      public Ideal(GenPolynomialRing<C> ring, List<GenPolynomial<C>> F, boolean gb, boolean topt)
      Constructor.
      Parameters:
      ring - polynomial ring
      F - list of polynomials
      gb - true if F is known to be a Groebner Base, else false
      topt - true if term order is optimized, else false

    • Ideal

      public Ideal(PolynomialList<C> list)
      Constructor.
      Parameters:
      list - polynomial list

    • Ideal

      public Ideal(PolynomialList<C> list, GroebnerBaseAbstract<C> bb, Reduction<C> red)
      Constructor.
      Parameters:
      list - polynomial list
      bb - Groebner Base engine
      red - Reduction engine

    • Ideal

      public Ideal(PolynomialList<C> list, boolean gb)
      Constructor.
      Parameters:
      list - polynomial list
      gb - true if list is known to be a Groebner Base, else false

    • Ideal

      public Ideal(PolynomialList<C> list, boolean gb, boolean topt)
      Constructor.
      Parameters:
      list - polynomial list
      gb - true if list is known to be a Groebner Base, else false
      topt - true if term order is optimized, else false

    • Ideal

      public Ideal(PolynomialList<C> list, boolean gb, GroebnerBaseAbstract<C> bb, Reduction<C> red)
      Constructor.
      Parameters:
      list - polynomial list
      gb - true if list is known to be a Groebner Base, else false
      bb - Groebner Base engine
      red - Reduction engine

    • Ideal

      public Ideal(PolynomialList<C> list, boolean gb, GroebnerBaseAbstract<C> bb)
      Constructor.
      Parameters:
      list - polynomial list
      gb - true if list is known to be a Groebner Base, else false
      bb - Groebner Base engine

    • Ideal

      public Ideal(PolynomialList<C> list, boolean gb, boolean topt, GroebnerBaseAbstract<C> bb)
      Constructor.
      Parameters:
      list - polynomial list
      gb - true if list is known to be a Groebner Base, else false
      topt - true if term order is optimized, else false
      bb - Groebner Base engine

    • Ideal

      public Ideal(PolynomialList<C> list, boolean gb, boolean topt, GroebnerBaseAbstract<C> bb, Reduction<C> red)
      Constructor.
      Parameters:
      list - polynomial list
      gb - true if list is known to be a Groebner Base, else false
      topt - true if term order is optimized, else false
      bb - Groebner Base engine
      red - Reduction engine

  • Method Details

    • copy

      public Ideal<C> copy()
      Clone this.
      Returns:
      a copy of this.

    • getList

      public List<GenPolynomial<C>> getList()
      Get the List of GenPolynomials.
      Returns:
      list.list

    • getRing

      public GenPolynomialRing<C> getRing()
      Get the GenPolynomialRing.
      Returns:
      list.ring

    • getZERO

      public Ideal<C> getZERO()
      Get the zero ideal.
      Returns:
      ideal(0)

    • getONE

      public Ideal<C> getONE()
      Get the one ideal.
      Returns:
      ideal(1)

    • toString

      public String toString()
      String representation of the ideal.
      Overrides:
      toString in class Object
      See Also:

    • toScript

      public String toScript()
      Get a scripting compatible string representation.
      Returns:
      script compatible representation for this Element.
      See Also:

    • equals

      public boolean equals(Object b)
      Comparison with any other object. Note: If both ideals are not Groebner Bases, then false may be returned even the ideals are equal.
      Overrides:
      equals in class Object
      See Also:

    • compareTo

      public int compareTo(Ideal<C> L)
      Ideal list comparison.
      Specified by:
      compareTo in interface Comparable<C extends GcdRingElem<C>>
      Parameters:
      L - other Ideal.
      Returns:
      compareTo() of polynomial lists.

    • hashCode

      public int hashCode()
      Hash code for this ideal.
      Overrides:
      hashCode in class Object
      See Also:

    • isZERO

      public boolean isZERO()
      Test if ZERO ideal.
      Returns:
      true, if this is the 0 ideal, else false

    • isONE

      public boolean isONE()
      Test if ONE is contained in the ideal. To test for a proper ideal use ! id.isONE().
      Returns:
      true, if this is the 1 ideal, else false

    • doToptimize

      public void doToptimize()
      Optimize the term order.

    • isGB

      public boolean isGB()
      Test if this is a Groebner base.
      Returns:
      true, if this is a Groebner base, else false

    • doGB

      public void doGB()
      Do Groebner Base. compute the Groebner Base for this ideal.

    • GB

      public Ideal<C> GB()
      Groebner Base. Get a Groebner Base for this ideal.
      Returns:
      GB(this)

    • contains

      public boolean contains(Ideal<C> B)
      Ideal containment. Test if B is contained in this ideal. Note: this is eventually modified to become a Groebner Base.
      Parameters:
      B - ideal
      Returns:
      true, if B is contained in this, else false

    • contains

      public boolean contains(GenPolynomial<C> b)
      Ideal containment. Test if b is contained in this ideal. Note: this is eventually modified to become a Groebner Base.
      Parameters:
      b - polynomial
      Returns:
      true, if b is contained in this, else false

    • contains

      public boolean contains(List<GenPolynomial<C>> B)
      Ideal containment. Test if each b in B is contained in this ideal. Note: this is eventually modified to become a Groebner Base.
      Parameters:
      B - list of polynomials
      Returns:
      true, if each b in B is contained in this, else false

    • sum

      public Ideal<C> sum(Ideal<C> B)
      Summation. Generators for the sum of ideals. Note: if both ideals are Groebner bases, a Groebner base is returned.
      Parameters:
      B - ideal
      Returns:
      ideal(this+B)

    • sum

      public Ideal<C> sum(GenPolynomial<C> b)
      Summation. Generators for the sum of ideal and a polynomial. Note: if this ideal is a Groebner base, a Groebner base is returned.
      Parameters:
      b - polynomial
      Returns:
      ideal(this+{b})

    • sum

      public Ideal<C> sum(List<GenPolynomial<C>> L)
      Summation. Generators for the sum of this ideal and a list of polynomials. Note: if this ideal is a Groebner base, a Groebner base is returned.
      Parameters:
      L - list of polynomials
      Returns:
      ideal(this+L)

    • product

      public Ideal<C> product(Ideal<C> B)
      Product. Generators for the product of ideals. Note: if both ideals are Groebner bases, a Groebner base is returned.
      Parameters:
      B - ideal
      Returns:
      ideal(this*B)

    • product

      public Ideal<C> product(GenPolynomial<C> b)
      Product. Generators for the product this ideal by a polynomial. Note: if this ideal is a Groebner base, a Groebner base is returned.
      Parameters:
      b - polynomial
      Returns:
      ideal(this*b)

    • intersect

      public Ideal<C> intersect(List<Ideal<C>> Bl)
      Intersection. Generators for the intersection of ideals. Using an iterative algorithm.
      Parameters:
      Bl - list of ideals
      Returns:
      ideal(cap_i B_i), a Groebner base

    • intersect

      public Ideal<C> intersect(Ideal<C> B)
      Intersection. Generators for the intersection of ideals.
      Parameters:
      B - ideal
      Returns:
      ideal(this \cap B), a Groebner base

    • intersect

      public Ideal<C> intersect(GenPolynomialRing<C> R)
      Intersection. Generators for the intersection of a ideal with a polynomial ring. The polynomial ring of this ideal must be a contraction of R and the TermOrder must be an elimination order.
      Parameters:
      R - polynomial ring
      Returns:
      ideal(this \cap R)

    • eliminate

      public Ideal<C> eliminate(GenPolynomialRing<C> R)
      Eliminate. Generators for the intersection of a ideal with a polynomial ring. The polynomial rings must have variable names.
      Parameters:
      R - polynomial ring
      Returns:
      ideal(this \cap R)

    • eliminate

      public Ideal<C> eliminate(String... ename)
      Eliminate. Preparation of generators for the intersection of a ideal with a polynomial ring.
      Parameters:
      ename - variables for the elimination ring.
      Returns:
      ideal(this) in K[ename,{vars \ ename}])

    • quotient

      public Ideal<C> quotient(GenPolynomial<C> h)
      Quotient. Generators for the ideal quotient.
      Parameters:
      h - polynomial
      Returns:
      ideal(this : h), a Groebner base

    • quotient

      public Ideal<C> quotient(Ideal<C> H)
      Quotient. Generators for the ideal quotient.
      Parameters:
      H - ideal
      Returns:
      ideal(this : H), a Groebner base

    • infiniteQuotientRab

      public Ideal<C> infiniteQuotientRab(GenPolynomial<C> h)
      Infinite quotient. Generators for the infinite ideal quotient.
      Parameters:
      h - polynomial
      Returns:
      ideal(this : hs), a Groebner base

    • infiniteQuotientExponent

      public int infiniteQuotientExponent(GenPolynomial<C> h, Ideal<C> Q)
      Infinite quotient exponent.
      Parameters:
      h - polynomial
      Q - quotient this : h^\infinity
      Returns:
      s with Q = this : hs

    • infiniteQuotient

      public Ideal<C> infiniteQuotient(GenPolynomial<C> h)
      Infinite quotient. Generators for the infinite ideal quotient.
      Parameters:
      h - polynomial
      Returns:
      ideal(this : hs), a Groebner base

    • isRadicalMember

      public boolean isRadicalMember(GenPolynomial<C> h)
      Radical membership test.
      Parameters:
      h - polynomial
      Returns:
      true if h is contained in the radical of ideal(this), else false.

    • infiniteQuotientOld

      public Ideal<C> infiniteQuotientOld(GenPolynomial<C> h)
      Infinite quotient. Generators for the infinite ideal quotient.
      Parameters:
      h - polynomial
      Returns:
      ideal(this : hs), a Groebner base

    • infiniteQuotient

      public Ideal<C> infiniteQuotient(Ideal<C> H)
      Infinite Quotient. Generators for the ideal infinite quotient.
      Parameters:
      H - ideal
      Returns:
      ideal(this : Hs), a Groebner base

    • infiniteQuotientRab

      public Ideal<C> infiniteQuotientRab(Ideal<C> H)
      Infinite Quotient. Generators for the ideal infinite quotient.
      Parameters:
      H - ideal
      Returns:
      ideal(this : Hs), a Groebner base

    • power

      public Ideal<C> power(int d)
      Power. Generators for the power of this ideal. Note: if this ideal is a Groebner base, a Groebner base is returned.
      Parameters:
      d - integer
      Returns:
      ideal(this^d)

    • normalform

      public GenPolynomial<C> normalform(GenPolynomial<C> h)
      Normalform for element.
      Parameters:
      h - polynomial
      Returns:
      normalform of h with respect to this

    • normalform

      public List<GenPolynomial<C>> normalform(List<GenPolynomial<C>> L)
      Normalform for list of elements.
      Parameters:
      L - polynomial list
      Returns:
      list of normalforms of the elements of L with respect to this

    • annihilator

      public Ideal<C> annihilator(GenPolynomial<C> h)
      Annihilator for element modulo this ideal.
      Parameters:
      h - polynomial
      Returns:
      annihilator of h with respect to this

    • isAnnihilator

      public boolean isAnnihilator(GenPolynomial<C> h, Ideal<C> A)
      Test for annihilator of element modulo this ideal.
      Parameters:
      h - polynomial
      A - ideal
      Returns:
      true, if A is the annihilator of h with respect to this

    • annihilator

      public Ideal<C> annihilator(Ideal<C> H)
      Annihilator for ideal modulo this ideal.
      Parameters:
      H - ideal
      Returns:
      annihilator of H with respect to this

    • isAnnihilator

      public boolean isAnnihilator(Ideal<C> H, Ideal<C> A)
      Test for annihilator of ideal modulo this ideal.
      Parameters:
      H - ideal
      A - ideal
      Returns:
      true, if A is the annihilator of H with respect to this

    • inverse

      public GenPolynomial<C> inverse(GenPolynomial<C> h)
      Inverse for element modulo this ideal.
      Parameters:
      h - polynomial
      Returns:
      inverse of h with respect to this, if defined

    • isUnit

      public boolean isUnit(GenPolynomial<C> h)
      Test if element is a unit modulo this ideal.
      Parameters:
      h - polynomial
      Returns:
      true if h is a unit with respect to this, else false

    • squarefree

      public Ideal<C> squarefree()
      Radical approximation. Squarefree generators for the ideal.
      Returns:
      squarefree(this), a Groebner base

    • commonZeroTest

      public int commonZeroTest()
      Ideal common zero test.
      Returns:
      -1, 0 or 1 if dimension(this) &eq; -1, 0 or ≥ 1.

    • isMaximal

      public boolean isMaximal()
      Test if this ideal is maximal.
      Returns:
      true, if this is certainly maximal and not one, else false.

    • univariateDegrees

      public List<Long> univariateDegrees()
      Univariate head term degrees.
      Returns:
      a list of the degrees of univariate head terms.

    • dimension

      public Dimension dimension()
      Ideal dimension.
      Returns:
      a dimension container (dim,maxIndep,list(maxIndep),vars).

    • dimension

      protected Set<Set<Integer>> dimension(Set<Integer> S, Set<Integer> U, Set<Set<Integer>> M)
      Ideal dimension.
      Parameters:
      S - is a set of independent variables.
      U - is a set of variables of unknown status.
      M - is a list of maximal sets of independent variables.
      Returns:
      a list of maximal sets of independent variables, eventually containing S.

    • containsHT

      protected boolean containsHT(Set<Integer> H, List<GenPolynomial<C>> G)
      Ideal head term containment test.
      Parameters:
      H - index set.
      G - list of polynomials.
      Returns:
      true, if the variables of the head terms of each polynomial in G are contained in H, else false.

    • contains

      protected boolean contains(int[] v, Set<Integer> H)
      Set containment. is v \subset H.
      Parameters:
      v - index array.
      H - index set.
      Returns:
      true, if each element of v is contained in H, else false .

    • constructUnivariate

      public List<GenPolynomial<C>> constructUnivariate()
      Construct univariate polynomials of minimal degree in all variables in zero dimensional ideal(G).
      Returns:
      list of univariate polynomial of minimal degree in each variable in ideal(G)

    • constructUnivariate

      public GenPolynomial<C> constructUnivariate(int i)
      Construct univariate polynomial of minimal degree in variable i in zero dimensional ideal(G).
      Parameters:
      i - variable index.
      Returns:
      univariate polynomial of minimal degree in variable i in ideal(G)

    • zeroDimRadicalDecomposition

      public List<IdealWithUniv<C>> zeroDimRadicalDecomposition()
      Zero dimensional radical decomposition. See Seidenbergs lemma 92, and BWK lemma 8.13.
      Returns:
      intersection of radical ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) )

    • isZeroDimRadical

      public boolean isZeroDimRadical()
      Test for Zero dimensional radical. See Seidenbergs lemma 92, and BWK lemma 8.13.
      Returns:
      true if this is an zero dimensional radical ideal, else false

    • isRadical

      public boolean isRadical(IdealWithUniv<C> ru)
      Test for radical ideal.
      Parameters:
      ru - ideal with univariate polynomials
      Returns:
      true if ru is a radical ideal, else false

    • zeroDimDecomposition

      public List<IdealWithUniv<C>> zeroDimDecomposition()
      Zero dimensional ideal irreducible decomposition. See algorithm DIRGZD of BGK 1986 and also PREDEC of the Gröbner bases book 1993.
      Returns:
      intersection H, of ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) ) and each ideal G_i has only irreducible minimal univariate polynomials and the G_i are pairwise co-prime.

    • zeroDimDecompositionExtension

      public List<IdealWithUniv<C>> zeroDimDecompositionExtension(List<GenPolynomial<C>> upol, List<GenPolynomial<C>> og)
      Zero dimensional ideal irreducible decomposition extension. One step decomposition via a minimal univariate polynomial in the lowest variable, used after each normalPosition step.
      Parameters:
      upol - list of univariate polynomials
      og - list of other generators for the ideal
      Returns:
      intersection of ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) ) and all minimal univariate polynomials of all G_i are irreducible

    • isZeroDimDecomposition

      public boolean isZeroDimDecomposition(List<IdealWithUniv<C>> L)
      Test for zero dimensional ideal decomposition.
      Parameters:
      L - intersection of ideals G_i with ideal(G) subseteq cap_i( ideal(G_i) ) and all minimal univariate polynomials of all G_i are irreducible
      Returns:
      true if L is a zero dimensional irreducible decomposition of this, else false

    • normalPositionFor

      public IdealWithUniv<C> normalPositionFor(int i, int j, List<GenPolynomial<C>> og)
      Compute normal position for variables i and j.
      Parameters:
      i - first variable index
      j - second variable index
      og - other generators for the ideal
      Returns:
      this + (z - x_j - t x_i) in the ring C[z, x_1, ..., x_r]

    • normalPositionForChar0

      IdealWithUniv<C> normalPositionForChar0(int i, int j, List<GenPolynomial<C>> og)
      Compute normal position for variables i and j, characteristic zero.
      Parameters:
      i - first variable index
      j - second variable index
      og - other generators for the ideal
      Returns:
      this + (z - x_j - t x_i) in the ring C[z, x_1, ..., x_r]

    • normalPositionForCharP

      IdealWithUniv<C> normalPositionForCharP(int i, int j, List<GenPolynomial<C>> og)
      Compute normal position for variables i and j, positive characteristic.
      Parameters:
      i - first variable index
      j - second variable index
      og - other generators for the ideal
      Returns:
      this + (z - x_j - t x_i) in the ring C[z, x_1, ..., x_r]

    • isNormalPositionFor

      public boolean isNormalPositionFor(int i, int j)
      Test if this ideal is in normal position for variables i and j.
      Parameters:
      i - first variable index
      j - second variable index
      Returns:
      true if this is in normal position with respect to i and j

    • normalPositionIndex2Vars

      public int[] normalPositionIndex2Vars()
      Normal position index, separate for polynomials with more than 2 variables. See also mas.masring.DIPDEC0#DIGISR
      Returns:
      (i,j) for non-normal variables

    • normalPositionIndexUnivars

      public int[] normalPositionIndexUnivars()
      Normal position index, separate multiple univariate polynomials. See also mas.masring.DIPDEC0#DIGISM
      Returns:
      (i,j) for non-normal variables

    • zeroDimRootDecomposition

      public List<IdealWithUniv<C>> zeroDimRootDecomposition()
      Zero dimensional ideal decomposition for real roots. See algorithm mas.masring.DIPDEC0#DINTSR.
      Returns:
      intersection of ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) ) and each G_i contains at most bi-variate polynomials and all univariate minimal polynomials are irreducible

    • zeroDimPrimeDecomposition

      public List<IdealWithUniv<C>> zeroDimPrimeDecomposition()
      Zero dimensional ideal prime decomposition. See algorithm mas.masring.DIPDEC0#DINTSS.
      Returns:
      intersection of ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) ) and each G_i is a prime ideal

    • zeroDimPrimeDecompositionFE

      public List<IdealWithUniv<C>> zeroDimPrimeDecompositionFE()
      Zero dimensional ideal prime decomposition, with field extension. See algorithm mas.masring.DIPDEC0#DINTSS.
      Returns:
      intersection of ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) ) and each G_i is a prime ideal with eventually containing field extension variables

    • primaryIdeal

      public Ideal<C> primaryIdeal(Ideal<C> P)
      Zero dimensional ideal associated primary ideal. See algorithm mas.masring.DIPIDEAL#DIRLPI.
      Parameters:
      P - prime ideal associated to this
      Returns:
      primary ideal of this with respect to the associated pime ideal P

    • zeroDimPrimaryDecomposition

      public List<PrimaryComponent<C>> zeroDimPrimaryDecomposition()
      Zero dimensional ideal primary decomposition.
      Returns:
      list of primary components of primary ideals G_i (pairwise co-prime) with ideal(this) = cap_i( ideal(G_i) ) together with the associated primes

    • zeroDimElimination

      public List<IdealWithUniv<C>> zeroDimElimination(List<IdealWithUniv<C>> pdec)
      Zero dimensional ideal elimination to original ring.
      Parameters:
      pdec - list of prime ideals G_i
      Returns:
      intersection of pairwise co-prime prime ideals G_i in the ring of this with ideal(this) = cap_i( ideal(G_i) )

    • zeroDimPrimaryDecomposition

      public List<PrimaryComponent<C>> zeroDimPrimaryDecomposition(List<IdealWithUniv<C>> pdec)
      Zero dimensional ideal primary decomposition.
      Parameters:
      pdec - list of prime ideals G_i with no field extensions
      Returns:
      list of primary components of primary ideals G_i (pairwise co-prime) with ideal(this) = cap_i( ideal(G_i) ) together with the associated primes

    • isPrimaryDecomposition

      public boolean isPrimaryDecomposition(List<PrimaryComponent<C>> L)
      Test for primary ideal decomposition.
      Parameters:
      L - list of primary components G_i
      Returns:
      true if ideal(this) == cap_i( ideal(G_i) )

    • extension

      public IdealWithUniv<Quotient<C>> extension(String... vars)
      Ideal extension.
      Parameters:
      vars - list of variables for a polynomial ring for extension
      Returns:
      ideal G, with coefficients in QuotientRing(GenPolynomialRing (vars))

    • extension

      public IdealWithUniv<Quotient<C>> extension(GenPolynomialRing<C> efac)
      Ideal extension.
      Parameters:
      efac - polynomial ring for extension
      Returns:
      ideal G, with coefficients in QuotientRing(efac)

    • extension

      public IdealWithUniv<Quotient<C>> extension(QuotientRing<C> qfac)
      Ideal extension.
      Parameters:
      qfac - quotient polynomial ring for extension
      Returns:
      ideal G, with coefficients in qfac

    • permContraction

      public IdealWithUniv<C> permContraction(IdealWithUniv<Quotient<C>> eideal)
      Ideal contraction and permutation.
      Parameters:
      eideal - extension ideal of this.
      Returns:
      contraction ideal of eideal in this polynomial ring

    • contraction

      public static <C extends GcdRingElem<C>> IdealWithUniv<C> contraction(IdealWithUniv<Quotient<C>> eid)
      Ideal contraction.
      Parameters:
      eid - extension ideal of this.
      Returns:
      contraction ideal of eid in distributed polynomial ring

    • permutation

      public static <C extends GcdRingElem<C>> IdealWithUniv<C> permutation(GenPolynomialRing<C> oring, IdealWithUniv<C> Cont)
      Ideal permutation.
      Parameters:
      oring - polynomial ring to which variables are back permuted.
      Cont - ideal to be permuted
      Returns:
      permutation of cont in polynomial ring oring

    • radical

      public Ideal<C> radical()
      Ideal radical.
      Returns:
      the radical ideal of this

    • radicalDecomposition

      public List<IdealWithUniv<C>> radicalDecomposition()
      Ideal radical decomposition.
      Returns:
      intersection of ideals G_i with radical(this) eq cap_i( ideal(G_i) ) and each G_i is a radical ideal and the G_i are pairwise co-prime

    • decomposition

      public List<IdealWithUniv<C>> decomposition()
      Ideal irreducible decomposition.
      Returns:
      intersection of ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) ) and each G_i is an ideal with irreducible univariate polynomials (after extension to a zero dimensional ideal) and the G_i are pairwise co-prime

    • primeDecomposition

      public List<IdealWithUniv<C>> primeDecomposition()
      Ideal prime decomposition.
      Returns:
      intersection of ideals G_i with ideal(this) subseteq cap_i( ideal(G_i) ) and each G_i is a prime ideal and the G_i are pairwise co-prime

    • isDecomposition

      public boolean isDecomposition(List<IdealWithUniv<C>> L)
      Test for ideal decomposition.
      Parameters:
      L - intersection of ideals G_i with ideal(G) eq cap_i(ideal(G_i) )
      Returns:
      true if L is a decomposition of this, else false

    • primaryDecomposition

      public List<PrimaryComponent<C>> primaryDecomposition()
      Ideal primary decomposition.
      Returns:
      list of primary components of primary ideals G_i (pairwise co-prime) with ideal(this) = cap_i( ideal(G_i) ) together with the associated primes