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

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

All Implemented Interfaces:

java.io.Serializable, java.lang.Cloneable, java.lang.Comparable<Ideal<C>>

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

Ideal implements some methods for ideal arithmetic, for example intersection, quotient and zero and positive dimensional ideal decomposition.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

protected GroebnerBaseAbstract<C>	bb Groebner base engine.
protected SquarefreeAbstract<C>	engine Squarefree decomposition engine.
protected boolean	isGB Indicator if list is a Groebner Base.
protected boolean	isTopt Indicator if list has optimized term order.
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)
Constructor.

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

Ideal(PolynomialList<C> list)
Constructor.

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

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

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

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

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

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

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

Method Summary

Ideal<C>	clone() Clone this.
int	commonZeroTest() Ideal common zero test.
int	compareTo(Ideal<C> L) Ideal list comparison.
java.util.List<GenPolynomial<C>>	constructUnivariate() Construct univariate polynomials of minimal degree in all variables in zero dimensional ideal(G).
GenPolynomial<C>	constructUnivariate(int i) Construct univariate polynomial of minimal degree in variable i in zero dimensional ideal(G).
GenPolynomial<C>	constructUnivariate(int i, java.util.List<GenPolynomial<C>> G) Construct univariate polynomial of minimal degree in variable i of a zero dimensional ideal(G).
boolean	contains(GenPolynomial<C> b) Ideal containment.
boolean	contains(Ideal<C> B) Ideal containment.
protected boolean	contains(int[] v, java.util.Set<java.lang.Integer> H) Set containment. is v \subset H.
boolean	contains(java.util.List<GenPolynomial<C>> B) Ideal containment.
protected boolean	containsHT(java.util.Set<java.lang.Integer> H, java.util.List<GenPolynomial<C>> G) Ideal head term containment test.
static <C extends GcdRingElem<C>> IdealWithUniv<C>	contraction(IdealWithUniv<Quotient<C>> eid) Ideal contraction.
java.util.List<IdealWithUniv<C>>	decomposition() Ideal irreducible decompostition.
Dimension	dimension() Ideal dimension.
protected java.util.Set<java.util.Set<java.lang.Integer>>	dimension(java.util.Set<java.lang.Integer> S, java.util.Set<java.lang.Integer> U, java.util.Set<java.util.Set<java.lang.Integer>> M) Ideal dimension.
void	doGB() Do Groebner Base. compute the Groebner Base for this ideal.
void	doToptimize() Optimize the term order.
Ideal<C>	eliminate(GenPolynomialRing<C> R) Eliminate.
Ideal<C>	eliminate(java.lang.String... ename) Eliminate.
boolean	equals(java.lang.Object b) Comparison with any other object.
IdealWithUniv<Quotient<C>>	extension(GenPolynomialRing<C> efac) Ideal extension.
IdealWithUniv<Quotient<C>>	extension(QuotientRing<C> qfac) Ideal extension.
IdealWithUniv<Quotient<C>>	extension(java.lang.String... vars) Ideal extension.
Ideal<C>	GB() Groebner Base.
java.util.List<GenPolynomial<C>>	getList() Get the List of GenPolynomials.
Ideal<C>	getONE() Get the one ideal.
GenPolynomialRing<C>	getRing() Get the GenPolynomialRing.
Ideal<C>	getZERO() Get the zero ideal.
int	hashCode() Hash code for this ideal.
Ideal<C>	infiniteQuotient(GenPolynomial<C> h) Infinite quotient.
Ideal<C>	infiniteQuotient(Ideal<C> H) Infinite Quotient.
int	infiniteQuotientExponent(GenPolynomial<C> h, Ideal<C> Q) Infinite quotient exponent.
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.
Ideal<C>	intersect(java.util.List<Ideal<C>> Bl) Intersection.
GenPolynomial<C>	inverse(GenPolynomial<C> h) Inverse for element modulo this ideal.
boolean	isDecomposition(java.util.List<IdealWithUniv<C>> L) Test for ideal decompostition.
boolean	isGB() Test if this is a Groebner base.
boolean	isMaximal() Test if this ideal is maximal.
boolean	isNormalPositionFor(int i, int j) Test if this ideal is in normal position for variables i and j.
boolean	isONE() Test if ONE is contained in the ideal.
boolean	isPrimaryDecomposition(java.util.List<PrimaryComponent<C>> L) Test for primary ideal decompostition.
boolean	isRadicalMember(GenPolynomial<C> h) Radical membership test.
boolean	isUnit(GenPolynomial<C> h) Test if element is a unit modulo this ideal.
boolean	isZERO() Test if ZERO ideal.
boolean	isZeroDimDecomposition(java.util.List<IdealWithUniv<C>> L) Test for zero dimensional ideal decompostition.
boolean	isZeroDimRadical() Test for Zero dimensional radical.
GenPolynomial<C>	normalform(GenPolynomial<C> h) Normalform for element.
java.util.List<GenPolynomial<C>>	normalform(java.util.List<GenPolynomial<C>> L) Normalform for list of elements.
IdealWithUniv<C>	normalPositionFor(int i, int j, java.util.List<GenPolynomial<C>> og) Compute normal position for variables i and j.
int[]	normalPositionIndex2Vars() Normal position index, separate for polynomials with more than 2 variables.
int[]	normalPositionIndexUnivars() Normal position index, separate multiple univariate polynomials.
IdealWithUniv<C>	permContraction(IdealWithUniv<Quotient<C>> eideal) Ideal contraction and permutation.
static <C extends GcdRingElem<C>> IdealWithUniv<C>	permutation(GenPolynomialRing<C> oring, IdealWithUniv<C> Cont) Ideal permutation.
Ideal<C>	power(int d) Power.
java.util.List<PrimaryComponent<C>>	primaryDecomposition() Ideal primary decompostition.
Ideal<C>	primaryIdeal(Ideal<C> P) Zero dimensional ideal associated primary ideal.
java.util.List<IdealWithUniv<C>>	primeDecomposition() Ideal prime decompostition.
Ideal<C>	product(Ideal<C> B) Product.
Ideal<C>	quotient(GenPolynomial<C> h) Quotient.
Ideal<C>	quotient(Ideal<C> H) Quotient.
Ideal<C>	radical() Ideal radical.
java.util.List<IdealWithUniv<C>>	radicalDecomposition() Ideal radical decompostition.
Ideal<C>	squarefree() Radical approximation.
Ideal<C>	sum(GenPolynomial<C> b) Summation.
Ideal<C>	sum(Ideal<C> B) Summation.
Ideal<C>	sum(java.util.List<GenPolynomial<C>> L) Summation.
java.lang.String	toScript() Get a scripting compatible string representation.
java.lang.String	toString() String representation of the ideal.
java.util.List<java.lang.Long>	univariateDegrees() Univariate head term degrees.
java.util.List<IdealWithUniv<C>>	zeroDimDecomposition() Zero dimensional ideal irreducible decompostition.
java.util.List<IdealWithUniv<C>>	zeroDimDecompositionExtension(java.util.List<GenPolynomial<C>> upol, java.util.List<GenPolynomial<C>> og) Zero dimensional ideal irreducible decompostition extension.
java.util.List<IdealWithUniv<C>>	zeroDimElimination(java.util.List<IdealWithUniv<C>> pdec) Zero dimensional ideal elimination to original ring.
java.util.List<PrimaryComponent<C>>	zeroDimPrimaryDecomposition() Zero dimensional ideal primary decompostition.
java.util.List<PrimaryComponent<C>>	zeroDimPrimaryDecomposition(java.util.List<IdealWithUniv<C>> pdec) Zero dimensional ideal primary decompostition.
java.util.List<IdealWithUniv<C>>	zeroDimPrimeDecomposition() Zero dimensional ideal prime decompostition.
java.util.List<IdealWithUniv<C>>	zeroDimPrimeDecompositionFE() Zero dimensional ideal prime decompostition, with field extension.
java.util.List<IdealWithUniv<C>>	zeroDimRadicalDecomposition() Zero dimensional radical decompostition.
java.util.List<IdealWithUniv<C>>	zeroDimRootDecomposition() Zero dimensional ideal decompostition for real roots.

Methods inherited from class java.lang.Object

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

Field Detail

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 Detail

Ideal

public Ideal(GenPolynomialRing<C> ring)

Constructor.

Parameters:

ring - polynomial ring

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(GenPolynomialRing<C> ring,
             java.util.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 Detail

clone

public Ideal<C> clone()

Clone this.

Overrides:

clone in class java.lang.Object

Returns:

a copy of this.

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

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 java.lang.String toString()

String representation of the 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 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)

compareTo

public int compareTo(Ideal<C> L)

Ideal list comparison.

Specified by:

compareTo in interface java.lang.Comparable<Ideal<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 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

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(java.util.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(java.util.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)

intersect

public Ideal<C> intersect(java.util.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(java.lang.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 : h^s), 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 : h^s

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

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 : 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

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 java.util.List<GenPolynomial<C>> normalform(java.util.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

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 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.

dimension

public Dimension dimension()

Ideal dimension.

Returns:

a dimension container (dim,maxIndep,list(maxIndep),vars).

dimension

protected java.util.Set<java.util.Set<java.lang.Integer>> dimension(java.util.Set<java.lang.Integer> S,
                                                                    java.util.Set<java.lang.Integer> U,
                                                                    java.util.Set<java.util.Set<java.lang.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(java.util.Set<java.lang.Integer> H,
                             java.util.List<GenPolynomial<C>> G)

Ideal head term containment test.

Parameters:

G - list of polynomials.

H - index set.

Returns:

true, if the vaiables of the head terms of each polynomial in G are contained in H, else false.

contains

protected boolean contains(int[] v,
                           java.util.Set<java.lang.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 java.util.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)

constructUnivariate

public GenPolynomial<C> constructUnivariate(int i,
                                            java.util.List<GenPolynomial<C>> G)

Construct univariate polynomial of minimal degree in variable i of a zero dimensional ideal(G).

Parameters:

i - variable index.

G - list of polynomials, a monic reduced Gröbner base of a zero dimensional ideal.

Returns:

univariate polynomial of minimal degree in variable i in ideal(G)

zeroDimRadicalDecomposition

public java.util.List<IdealWithUniv<C>> zeroDimRadicalDecomposition()

Zero dimensional radical decompostition. 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

zeroDimDecomposition

public java.util.List<IdealWithUniv<C>> zeroDimDecomposition()

Zero dimensional ideal irreducible decompostition. 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 java.util.List<IdealWithUniv<C>> zeroDimDecompositionExtension(java.util.List<GenPolynomial<C>> upol,
                                                                      java.util.List<GenPolynomial<C>> og)

Zero dimensional ideal irreducible decompostition 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(java.util.List<IdealWithUniv<C>> L)

Test for zero dimensional ideal decompostition.

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,
                                          java.util.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]

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 java.util.List<IdealWithUniv<C>> zeroDimRootDecomposition()

Zero dimensional ideal decompostition 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 java.util.List<IdealWithUniv<C>> zeroDimPrimeDecomposition()

Zero dimensional ideal prime decompostition. 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 java.util.List<IdealWithUniv<C>> zeroDimPrimeDecompositionFE()

Zero dimensional ideal prime decompostition, 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 java.util.List<PrimaryComponent<C>> zeroDimPrimaryDecomposition()

Zero dimensional ideal primary decompostition.

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 java.util.List<IdealWithUniv<C>> zeroDimElimination(java.util.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 java.util.List<PrimaryComponent<C>> zeroDimPrimaryDecomposition(java.util.List<IdealWithUniv<C>> pdec)

Zero dimensional ideal primary decompostition.

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(java.util.List<PrimaryComponent<C>> L)

Test for primary ideal decompostition.

Parameters:

L - list of primary components G_i

Returns:

true if ideal(this) == cap_i( ideal(G_i) )

extension

public IdealWithUniv<Quotient<C>> extension(java.lang.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 java.util.List<IdealWithUniv<C>> radicalDecomposition()

Ideal radical decompostition.

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 java.util.List<IdealWithUniv<C>> decomposition()

Ideal irreducible decompostition.

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 java.util.List<IdealWithUniv<C>> primeDecomposition()

Ideal prime decompostition.

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(java.util.List<IdealWithUniv<C>> L)

Test for ideal decompostition.

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 java.util.List<PrimaryComponent<C>> primaryDecomposition()

Ideal primary decompostition.

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