edu.jas.application
Class Ideal<C extends RingElem<C>>

java.lang.Object
  edu.jas.application.Ideal<C>

All Implemented Interfaces:

java.io.Serializable

public class Ideal<C extends RingElem<C>>
extends java.lang.Object
implements java.io.Serializable

Ideal implements some methods for ideal arithmetic, e.g. intersection and quotient.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

protected GroebnerBase<C>	bb Groebner base engine.
protected boolean	isGB Indicator if list is a Groebner Base.
protected PolynomialList<C>	list The data structure is a PolynomialList.
protected Reduction<C>	red Reduction engine.
protected boolean	testGB Indicator if test has been performed if this is a Groebner Base.

Constructor Summary

Ideal(GenPolynomialRing<C> ring, java.util.List<GenPolynomial<C>> F)
Constructor.

Ideal(GenPolynomialRing<C> ring, java.util.List<GenPolynomial<C>> F, boolean gb)
Constructor.

Ideal(PolynomialList<C> list)
Constructor.

Ideal(PolynomialList<C> list, boolean gb)
Constructor.

Ideal(PolynomialList<C> list, boolean gb, GroebnerBase<C> bb, Reduction<C> red)
Constructor.

Ideal(PolynomialList<C> list, GroebnerBase<C> bb, Reduction<C> red)
Constructor.

Method Summary

boolean	contains(Ideal<C> B) Ideal containment.
boolean	equals(java.lang.Object b) Comparison with any other object.
Ideal<C>	GB() Groebner Base.
java.util.List<GenPolynomial<C>>	getList() Get the List of GenPolynomials.
GenPolynomialRing<C>	getRing() Get the GenPolynomialRing.
int	hashCode() Hash code for this ideal.
Ideal<C>	infiniteQuotient(GenPolynomial<C> h) Infinite quotient.
Ideal<C>	infiniteQuotient(Ideal<C> H) Infinite Quotient.
Ideal<C>	infiniteQuotientOld(GenPolynomial<C> h) Infinite quotient.
Ideal<C>	infiniteQuotientRab(GenPolynomial<C> h) Infinite quotient.
Ideal<C>	infiniteQuotientRab(Ideal<C> H) Infinite Quotient.
Ideal<C>	intersect(GenPolynomialRing<C> R) Intersection.
Ideal<C>	intersect(Ideal<C> B) Intersection.
GenPolynomial<C>	inverse(GenPolynomial<C> h) Inverse for element modulo this ideal.
boolean	isGB() Test if this is a Groebner base.
boolean	isONE() Test if ONE ideal.
boolean	isUnit(GenPolynomial<C> h) Test if element is a unit modulo this ideal.
boolean	isZERO() Test if ZERO ideal.
GenPolynomial<C>	normalform(GenPolynomial<C> h) Normalform for element.
Ideal<C>	product(Ideal<C> B) Product.
Ideal<C>	quotient(GenPolynomial<C> h) Quotient.
Ideal<C>	quotient(Ideal<C> H) Quotient.
Ideal<C>	sum(Ideal<C> B) Summation.
java.lang.String	toString() String representation of the ideal.

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Detail

list

protected PolynomialList<C extends RingElem<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.

bb

protected GroebnerBase<C extends RingElem<C>> bb

Groebner base engine.

red

protected Reduction<C extends RingElem<C>> red

Reduction engine.

Constructor Detail

Ideal

public Ideal(GenPolynomialRing<C> ring,
             java.util.List<GenPolynomial<C>> F)

Constructor.

Parameters:

ring - polynomial ring

F - list of polynomials

Ideal

public Ideal(GenPolynomialRing<C> ring,
             java.util.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(PolynomialList<C> list)

Constructor.

Parameters:

list - polynomial list

Ideal

public Ideal(PolynomialList<C> list,
             GroebnerBase<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,
             GroebnerBase<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

Method Detail

getList

public java.util.List<GenPolynomial<C>> getList()

Get the List of GenPolynomials.

Returns:

list.list

getRing

public GenPolynomialRing<C> getRing()

Get the GenPolynomialRing.

Returns:

list.ring

toString

public java.lang.String toString()

String representation of the ideal.

Overrides:

toString in class java.lang.Object

See Also:

Object.toString()

equals

public boolean equals(java.lang.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 java.lang.Object

See Also:

Object.equals(java.lang.Object)

hashCode

public int hashCode()

Hash code for this ideal.

Overrides:

hashCode in class java.lang.Object

See Also:

Object.hashCode()

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 ideal.

Returns:

true, if this is the 1 ideal, else false

isGB

public boolean isGB()

Test if this is a Groebner base.

Returns:

true, if this is a Groebner base, else false

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

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)

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)

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)

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 : h^s), a Groebner base

infiniteQuotient

public Ideal<C> infiniteQuotient(GenPolynomial<C> h)

Infinite quotient. Generators for the infinite ideal quotient.

Parameters:

h - polynomial

Returns:

ideal(this : h^s), a Groebner base

infiniteQuotientOld

public Ideal<C> infiniteQuotientOld(GenPolynomial<C> h)

Infinite quotient. Generators for the infinite ideal quotient.

Parameters:

h - polynomial

Returns:

ideal(this : h^s), 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 : H^s), 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 : H^s), a Groebner base

normalform

public GenPolynomial<C> normalform(GenPolynomial<C> h)

Normalform for element.

Parameters:

h - polynomial

Returns:

normalform 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