edu.jas.application

Class WordIdeal<C extends GcdRingElem<C>>

java.lang.Object

edu.jas.application.WordIdeal<C>

All Implemented Interfaces:

java.io.Serializable, java.lang.Comparable<WordIdeal<C>>

public class WordIdeal<C extends GcdRingElem<C>>
extends java.lang.Object
implements java.lang.Comparable<WordIdeal<C>>, java.io.Serializable

Word Ideal implements some methods for ideal arithmetic, for example containment, sum or product. Note: only two-sided ideals.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
protected WordGroebnerBaseAbstract<C>	bb Groebner base engine.
protected boolean	isGB Indicator if list is a Groebner Base.
protected java.util.List<GenWordPolynomial<C>>	list The data structure is a list of word polynomials.
protected WordReduction<C>	red Reduction engine.
protected GenWordPolynomialRing<C>	ring Reference to the word polynomial ring.
protected boolean	testGB Indicator if test has been performed if this is a Groebner Base.

Constructor Summary

Constructors

Constructor and Description
WordIdeal(GenWordPolynomialRing<C> ring) Constructor.
WordIdeal(GenWordPolynomialRing<C> ring, java.util.List<GenWordPolynomial<C>> list) Constructor.
WordIdeal(GenWordPolynomialRing<C> ring, java.util.List<GenWordPolynomial<C>> list, boolean gb) Constructor.
WordIdeal(GenWordPolynomialRing<C> ring, java.util.List<GenWordPolynomial<C>> list, boolean gb, WordGroebnerBaseAbstract<C> bb) Constructor.
WordIdeal(GenWordPolynomialRing<C> ring, java.util.List<GenWordPolynomial<C>> list, boolean gb, WordGroebnerBaseAbstract<C> bb, WordReduction<C> red) Constructor.
WordIdeal(GenWordPolynomialRing<C> ring, java.util.List<GenWordPolynomial<C>> list, WordGroebnerBaseAbstract<C> bb, WordReduction<C> red) Constructor.

Method Summary

Modifier and Type	Method and Description
int	commonZeroTest() Ideal common zero test.
int	compareTo(WordIdeal<C> L) WordIdeal comparison.
boolean	contains(GenWordPolynomial<C> b) Word ideal containment.
boolean	contains(java.util.List<GenWordPolynomial<C>> B) Word ideal containment.
boolean	contains(WordIdeal<C> B) Word ideal containment.
WordIdeal<C>	copy() Clone this.
void	doGB() Do Groebner Base.
WordIdeal<C>	eliminate(GenWordPolynomialRing<C> R)
boolean	equals(java.lang.Object b) Comparison with any other object.
WordIdeal<C>	GB() Groebner Base.
java.util.List<GenWordPolynomial<C>>	getList() Get the List of GenWordPolynomials.
WordIdeal<C>	getONE() Get the one ideal.
GenWordPolynomialRing<C>	getRing() Get the GenWordPolynomialRing.
WordIdeal<C>	getZERO() Get the zero ideal.
int	hashCode() Hash code for this word ideal.
WordIdeal<C>	intersect(GenWordPolynomialRing<C> R)
WordIdeal<C>	intersect(java.util.List<WordIdeal<C>> Bl)
WordIdeal<C>	intersect(WordIdeal<C> B)
GenWordPolynomial<C>	inverse(GenWordPolynomial<C> h) Inverse for element modulo this ideal.
boolean	isGB() Test if this is a twosided Groebner base.
boolean	isMaximal() Test if this ideal is maximal.
boolean	isONE() Test if ONE is contained in the ideal.
boolean	isUnit(GenWordPolynomial<C> h) Test if element is a unit modulo this ideal.
boolean	isZERO() Test if ZERO ideal.
GenWordPolynomial<C>	normalform(GenWordPolynomial<C> h) Normalform for element.
java.util.List<GenWordPolynomial<C>>	normalform(java.util.List<GenWordPolynomial<C>> L) Normalform for list of word elements.
WordIdeal<C>	power(int d) Power.
WordIdeal<C>	product(GenWordPolynomial<C> b) Left product.
WordIdeal<C>	product(WordIdeal<C> B) Product.
WordIdeal<C>	sum(GenWordPolynomial<C> b) Word summation.
WordIdeal<C>	sum(java.util.List<GenWordPolynomial<C>> L) Word summation.
WordIdeal<C>	sum(WordIdeal<C> B) Word ideal summation.
java.lang.String	toScript() Get a scripting compatible string representation.
java.lang.String	toString() String representation of the word ideal.
java.util.List<java.lang.Long>	univariateDegrees() Univariate head term degrees.

Methods inherited from class java.lang.Object

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

Field Detail

list

protected java.util.List<GenWordPolynomial<C extends GcdRingElem<C>>> list

The data structure is a list of word polynomials.

ring

protected GenWordPolynomialRing<C extends GcdRingElem<C>> ring

Reference to the word polynomial ring.

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 final WordGroebnerBaseAbstract<C extends GcdRingElem<C>> bb

Groebner base engine.

red

protected final WordReduction<C extends GcdRingElem<C>> red

Reduction engine.

Constructor Detail

WordIdeal

public WordIdeal(GenWordPolynomialRing<C> ring)

Constructor.

Parameters:

ring - word polynomial ring

WordIdeal

public WordIdeal(GenWordPolynomialRing<C> ring,
                 java.util.List<GenWordPolynomial<C>> list)

Constructor.

Parameters:

ring - word polynomial ring

list - word polynomial list

WordIdeal

public WordIdeal(GenWordPolynomialRing<C> ring,
                 java.util.List<GenWordPolynomial<C>> list,
                 WordGroebnerBaseAbstract<C> bb,
                 WordReduction<C> red)

Constructor.

Parameters:

ring - word polynomial ring

list - word polynomial list

bb - Groebner Base engine

red - Reduction engine

WordIdeal

public WordIdeal(GenWordPolynomialRing<C> ring,
                 java.util.List<GenWordPolynomial<C>> list,
                 boolean gb)

Constructor.

Parameters:

ring - word polynomial ring

list - word polynomial list

gb - true if list is known to be a Groebner Base, else false

WordIdeal

public WordIdeal(GenWordPolynomialRing<C> ring,
                 java.util.List<GenWordPolynomial<C>> list,
                 boolean gb,
                 WordGroebnerBaseAbstract<C> bb)

Constructor.

Parameters:

ring - word polynomial ring

list - word polynomial list

gb - true if list is known to be a Groebner Base, else false

bb - Groebner Base engine

WordIdeal

public WordIdeal(GenWordPolynomialRing<C> ring,
                 java.util.List<GenWordPolynomial<C>> list,
                 boolean gb,
                 WordGroebnerBaseAbstract<C> bb,
                 WordReduction<C> red)

Constructor.

Parameters:

ring - word polynomial ring

list - word polynomial list

gb - true if list is known to be a Groebner Base, else false

bb - Groebner Base engine

red - Reduction engine

Method Detail

copy

public WordIdeal<C> copy()

Clone this.

Returns:

a copy of this.

getList

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

Get the List of GenWordPolynomials.

Returns:

(cast) list.list

getRing

public GenWordPolynomialRing<C> getRing()

Get the GenWordPolynomialRing.

Returns:

(cast) list.ring

getZERO

public WordIdeal<C> getZERO()

Get the zero ideal.

Returns:

ideal(0)

getONE

public WordIdeal<C> getONE()

Get the one ideal.

Returns:

ideal(1)

toString

public java.lang.String toString()

String representation of the word ideal.

Overrides:

toString in class java.lang.Object

See Also:

Object.toString()

toScript

public java.lang.String toScript()

Get a scripting compatible string representation.

Returns:

script compatible representation for this Element.

See Also:

Element.toScript()

equals

public boolean equals(java.lang.Object b)

Comparison with any other object. Note: If not both ideals are 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)

compareTo

public int compareTo(WordIdeal<C> L)

WordIdeal comparison.

Specified by:

compareTo in interface java.lang.Comparable<WordIdeal<C extends GcdRingElem<C>>>

Parameters:

L - other word ideal.

Returns:

compareTo() of polynomial lists.

hashCode

public int hashCode()

Hash code for this word 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 is contained in the ideal. To test for a proper ideal use ! id.isONE().

Returns:

true, if this is the 1 ideal, else false

isGB

public boolean isGB()

Test if this is a twosided Groebner base.

Returns:

true, if this is a twosided Groebner base, else false

doGB

public void doGB()

Do Groebner Base. Compute the Groebner Base for this ideal.

GB

public WordIdeal<C> GB()

Groebner Base. Get a Groebner Base for this ideal.

Returns:

twosidedGB(this)

contains

public boolean contains(WordIdeal<C> B)

Word ideal containment. Test if B is contained in this ideal. Note: this is eventually modified to become a Groebner Base.

Parameters:

B - word ideal

Returns:

true, if B is contained in this, else false

contains

public boolean contains(GenWordPolynomial<C> b)

Word ideal containment. Test if b is contained in this ideal. Note: this is eventually modified to become a Groebner Base.

Parameters:

b - word polynomial

Returns:

true, if b is contained in this, else false

contains

public boolean contains(java.util.List<GenWordPolynomial<C>> B)

Word 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 word polynomials

Returns:

true, if each b in B is contained in this, else false

sum

public WordIdeal<C> sum(WordIdeal<C> B)

Word ideal summation. Generators for the sum of ideals. Note: if both ideals are Groebner bases, a Groebner base is returned.

Parameters:

B - word ideal

Returns:

ideal(this+B)

sum

public WordIdeal<C> sum(GenWordPolynomial<C> b)

Word 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 - word polynomial

Returns:

ideal(this+{b})

sum

public WordIdeal<C> sum(java.util.List<GenWordPolynomial<C>> L)

Word 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 word polynomials

Returns:

ideal(this+L)

product

public WordIdeal<C> product(WordIdeal<C> B)

Product. Generators for the product of ideals. Note: if both ideals are Groebner bases, a Groebner base is returned.

Parameters:

B - word ideal

Returns:

ideal(this*B)

product

public WordIdeal<C> product(GenWordPolynomial<C> b)

Left product. Generators for the product this by a polynomial.

Parameters:

b - word polynomial

Returns:

ideal(this*b)

intersect

public WordIdeal<C> intersect(java.util.List<WordIdeal<C>> Bl)

intersect

public WordIdeal<C> intersect(WordIdeal<C> B)

intersect

public WordIdeal<C> intersect(GenWordPolynomialRing<C> R)

eliminate

public WordIdeal<C> eliminate(GenWordPolynomialRing<C> R)

power

public WordIdeal<C> power(int d)

Power. Generators for the power of this word ideal. Note: if this ideal is a Groebner base, a Groebner base is returned.

Parameters:

d - integer

Returns:

ideal(this^d)

normalform

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

Normalform for element.

Parameters:

h - word polynomial

Returns:

left normalform of h with respect to this

normalform

public java.util.List<GenWordPolynomial<C>> normalform(java.util.List<GenWordPolynomial<C>> L)

Normalform for list of word elements.

Parameters:

L - word polynomial list

Returns:

list of left normalforms of the elements of L with respect to this

inverse

public GenWordPolynomial<C> inverse(GenWordPolynomial<C> h)

Inverse for element modulo this ideal.

Parameters:

h - word polynomial

Returns:

inverse of h with respect to this, if defined

isUnit

public boolean isUnit(GenWordPolynomial<C> h)

Test if element is a unit modulo this ideal.

Parameters:

h - word polynomial

Returns:

true if h is a unit with respect to this, else false

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 maximal and not one, else false.

univariateDegrees

public java.util.List<java.lang.Long> univariateDegrees()

Univariate head term degrees.

Returns:

a list of the degrees of univariate head terms.