java.lang.Object
- cc.redberry.rings.poly.multivar.Ideal<Term,Poly>

All Implemented Interfaces:: Stringifiable<Poly>, Serializable

public final class Ideal<Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>>
extends Object
implements Stringifiable<Poly>, Serializable

Ideal represented by its Groebner basis.

Since:: 2.3
See Also:: Serialized Form

Field Summary

Fields
Modifier and Type Field Description

Comparator<DegreeVector> ordering
monomial order used for standard basis

Method Summary

Modifier and Type	Method	Description
`Ideal<Term,Poly>`	`changeOrder(Comparator<DegreeVector> newMonomialOrder)`	Set the monomial order used for Groebner basis of this ideal
`boolean`	`contains(Ideal<Term,Poly> oth)`	Whether this ideal contains the specified one
`boolean`	`contains(Poly poly)`	Tests whether specified poly is an element of this ideal
`boolean`	`containsProduct(Ideal<Term,Poly> a, Ideal<Term,Poly> b)`	Whether this ideal contains the product of two specified ideals
`static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly>`	`create(List<Poly> generators)`	Creates ideal given by a list of generators.
`static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly>`	`create(List<Poly> generators, Comparator<DegreeVector> monomialOrder)`	Creates ideal given by a list of generators.
`static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly>`	`create(Poly... generators)`	Creates ideal given by a list of generators.
`int`	`degree()`	Returns the affine degree of this ideal
`int`	`dimension()`	Returns the affine dimension of this ideal
`static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly>`	`empty(Poly factory)`	Creates empty ideal
`static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly>`	`empty(Poly factory, Comparator<DegreeVector> monomialOrder)`	Creates empty ideal
`boolean`	`equals(Object o)`
`Poly`	`getBasisGenerator(int i)`	Returns i-th element of Groebner basis
`List<Poly>`	`getGroebnerBasis()`	Groebner basis of this ideal
`Comparator<DegreeVector>`	`getMonomialOrder()`	The monomial order used for Groebner basis
`List<Poly>`	`getOriginalGenerators()`	Returns the list of original generators
`int`	`hashCode()`
`GroebnerBases.HilbertSeries`	`hilbertSeries()`	Hilbert-Poincare series of this ideal
`Ideal<Term,Poly>`	`intersection(Ideal<Term,Poly> oth)`	Returns the intersection of this and oth
`boolean`	`isEmpty()`	Whether this ideal is empty
`boolean`	`isHomogeneous()`	Whether this ideal is homogeneous
`boolean`	`isMaximal()`	Returns true if this ideal is maximal (that is its affine variety has only one point)
`boolean`	`isMonomial()`	Whether this ideal is monomial
`boolean`	`isPrincipal()`	Whether this ideal is principal
`boolean`	`isProper()`	Whether this is a proper ideal
`boolean`	`isTrivial()`	Whether this ideal is the whole ring (basis consists of pne constant polynomial)
`Ideal<Term,Poly>`	`ltIdeal()`	Ideal of leading terms
`Ideal<Term,Poly>`	`multiply(Ideal<Term,Poly> oth)`	Returns the product of this and oth
`Ideal<Term,Poly>`	`multiply(Poly oth)`	Returns the product of this and oth
`int`	`nBasisGenerators()`	Returns the number of elements in Groebner basis
`Poly`	`normalForm(Poly poly)`	Reduces `poly` modulo this ideal
`static <E> Ideal<Monomial<E>,MultivariatePolynomial<E>>`	`parse(String[] generators, Ring<E> field, String[] variables)`	Shortcut for parse
`static <E> Ideal<Monomial<E>,MultivariatePolynomial<E>>`	`parse(String[] generators, Ring<E> field, Comparator<DegreeVector> monomialOrder, String[] variables)`	Shortcut for parse
`Ideal<Term,Poly>`	`pow(int exponent)`	Returns this in a power of exponent
`Ideal<Term,Poly>`	`quotient(Ideal<Term,Poly> oth)`	Returns the quotient this : oth
`Ideal<Term,Poly>`	`quotient(Poly oth)`	Returns the quotient this : oth
`boolean`	`radicalContains(Poly poly)`	Tests whether `poly` belongs to the radical of this
`Ideal<Term,Poly>`	`square()`	Returns squared ideal
`String`	`toString()`
`String`	`toString(IStringifier<Poly> stringifier)`	convert this to string with the use of stringifier
`static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly>`	`trivial(Poly factory)`	Creates trivial ideal (ideal = ring)
`static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly>`	`trivial(Poly factory, Comparator<DegreeVector> monomialOrder)`	Creates trivial ideal (ideal = ring)
`Ideal<Term,Poly>`	`union(Ideal<Term,Poly> oth)`	Returns the union of this and oth
`Ideal<Term,Poly>`	`union(Poly oth)`	Returns the union of this and oth

Methods inherited from class java.lang.Object
clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Detail
- ordering
```
public final Comparator<DegreeVector> ordering
```
  monomial order used for standard basis

Method Detail

getMonomialOrder
```
public Comparator<DegreeVector> getMonomialOrder()
```
The monomial order used for Groebner basis

changeOrder

public Ideal<Term,Poly> changeOrder(Comparator<DegreeVector> newMonomialOrder)

Set the monomial order used for Groebner basis of this ideal

normalForm
```
public Poly normalForm(Poly poly)
```
Reduces poly modulo this ideal

getOriginalGenerators
```
public List<Poly> getOriginalGenerators()
```
Returns the list of original generators

getGroebnerBasis
```
public List<Poly> getGroebnerBasis()
```
Groebner basis of this ideal

nBasisGenerators
```
public int nBasisGenerators()
```
Returns the number of elements in Groebner basis

getBasisGenerator
```
public Poly getBasisGenerator(int i)
```
Returns i-th element of Groebner basis

isTrivial
```
public boolean isTrivial()
```
Whether this ideal is the whole ring (basis consists of pne constant polynomial)

isProper
```
public boolean isProper()
```
Whether this is a proper ideal

isEmpty
```
public boolean isEmpty()
```
Whether this ideal is empty

isPrincipal
```
public boolean isPrincipal()
```
Whether this ideal is principal

isHomogeneous
```
public boolean isHomogeneous()
```
Whether this ideal is homogeneous

isMonomial
```
public boolean isMonomial()
```
Whether this ideal is monomial

isMaximal
```
public boolean isMaximal()
```
Returns true if this ideal is maximal (that is its affine variety has only one point)

ltIdeal
```
public Ideal<Term,Poly> ltIdeal()
```
Ideal of leading terms

contains
```
public boolean contains(Poly poly)
```
Tests whether specified poly is an element of this ideal

contains

public boolean contains(Ideal<Term,Poly> oth)

Whether this ideal contains the specified one

hilbertSeries

public GroebnerBases.HilbertSeries hilbertSeries()

Hilbert-Poincare series of this ideal

dimension
```
public int dimension()
```
Returns the affine dimension of this ideal

degree
```
public int degree()
```
Returns the affine degree of this ideal

containsProduct

public boolean containsProduct(Ideal<Term,Poly> a,
                               Ideal<Term,Poly> b)

Whether this ideal contains the product of two specified ideals

radicalContains
```
public boolean radicalContains(Poly poly)
```
Tests whether poly belongs to the radical of this

union

public Ideal<Term,Poly> union(Poly oth)

Returns the union of this and oth

union

public Ideal<Term,Poly> union(Ideal<Term,Poly> oth)

Returns the union of this and oth

multiply

public Ideal<Term,Poly> multiply(Ideal<Term,Poly> oth)

Returns the product of this and oth

square
```
public Ideal<Term,Poly> square()
```
Returns squared ideal

pow

public Ideal<Term,Poly> pow(int exponent)

Returns this in a power of exponent

multiply

public Ideal<Term,Poly> multiply(Poly oth)

Returns the product of this and oth

intersection

public Ideal<Term,Poly> intersection(Ideal<Term,Poly> oth)

Returns the intersection of this and oth

quotient

public Ideal<Term,Poly> quotient(Poly oth)

Returns the quotient this : oth

quotient

public Ideal<Term,Poly> quotient(Ideal<Term,Poly> oth)

Returns the quotient this : oth

equals
```
public boolean equals(Object o)
```
Overrides:

equals in class Object

hashCode
```
public int hashCode()
```
Overrides:

hashCode in class Object

toString
```
public String toString(IStringifier<Poly> stringifier)
```
Description copied from interface: Stringifiable

convert this to string with the use of stringifier

Specified by:

toString in interface Stringifiable<Term extends AMonomial<Term>>

toString
```
public String toString()
```
Overrides:

toString in class Object

create

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly> create(List<Poly> generators)

Creates ideal given by a list of generators. Groebner basis with respect to GREVLEX order will be used.

create

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly> create(Poly... generators)

Creates ideal given by a list of generators. Groebner basis with respect to GREVLEX order will be used.

create

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly> create(List<Poly> generators,
                                                                                                                                       Comparator<DegreeVector> monomialOrder)

Creates ideal given by a list of generators. Groebner basis with respect to specified monomialOrder will be used.

Parameters:: monomialOrder - monomial order for unique Groebner basis of the ideal

trivial

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly> trivial(Poly factory)

Creates trivial ideal (ideal = ring)

trivial

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly> trivial(Poly factory,
                                                                                                                                        Comparator<DegreeVector> monomialOrder)

Creates trivial ideal (ideal = ring)

empty

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly> empty(Poly factory)

Creates empty ideal

empty

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> Ideal<Term,Poly> empty(Poly factory,
                                                                                                                                      Comparator<DegreeVector> monomialOrder)

Creates empty ideal

parse

public static <E> Ideal<Monomial<E>,MultivariatePolynomial<E>> parse(String[] generators,
                                                                           Ring<E> field,
                                                                           String[] variables)

Shortcut for parse

parse

public static <E> Ideal<Monomial<E>,MultivariatePolynomial<E>> parse(String[] generators,
                                                                           Ring<E> field,
                                                                           Comparator<DegreeVector> monomialOrder,
                                                                           String[] variables)

Shortcut for parse

Class Ideal<Term extends AMonomial<Term>,​Poly extends AMultivariatePolynomial<Term,​Poly>>

Field Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

ordering

Method Detail

getMonomialOrder

changeOrder

normalForm

getOriginalGenerators

getGroebnerBasis

nBasisGenerators

getBasisGenerator

isTrivial

isProper

isEmpty

isPrincipal

isHomogeneous

isMonomial

isMaximal

ltIdeal

contains

contains

hilbertSeries

dimension

degree

containsProduct

radicalContains

union

union

multiply

square

pow

multiply

intersection

quotient

quotient

equals

hashCode

toString

toString

create

create

create

trivial

trivial

empty

empty

parse

parse

Class Ideal<Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>>