edu.jas.gbmod

Class SolvableSyzygyAbstract<C extends RingElem<C>>

java.lang.Object

edu.jas.gbmod.SolvableSyzygyAbstract<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

SolvableSyzygy<C>, java.io.Serializable

public class SolvableSyzygyAbstract<C extends RingElem<C>>
extends java.lang.Object
implements SolvableSyzygy<C>

Syzygy class for solvable polynomials. Implements Syzygy computations and tests.

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
SolvableReduction<C>	sred Solvable reduction engine.

Constructor Summary

Constructors

Constructor and Description
SolvableSyzygyAbstract() Constructor.

Method Summary

Modifier and Type	Method and Description
int	compare(GenSolvablePolynomial<C> num, GenSolvablePolynomial<C> den, GenSolvablePolynomial<C> n, GenSolvablePolynomial<C> d) Comparison like SolvableLocal or SolvableQuotient.
boolean	isLeftOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b, GenSolvablePolynomial<C>[] oc) Test left Ore condition.
boolean	isLeftZeroRelation(java.util.List<java.util.List<GenSolvablePolynomial<C>>> Z, java.util.List<GenSolvablePolynomial<C>> F) Test if left syzygy.
boolean	isLeftZeroRelation(ModuleList<C> Z, ModuleList<C> F) Test if left sysygy of modules
boolean	isRightOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b, GenSolvablePolynomial<C>[] oc) Test right Ore condition.
boolean	isRightZeroRelation(java.util.List<java.util.List<GenSolvablePolynomial<C>>> Z, java.util.List<GenSolvablePolynomial<C>> F) Test if right syzygy.
boolean	isRightZeroRelation(ModuleList<C> Z, ModuleList<C> F) Test if right sysygy of modules
GenSolvablePolynomial<C>[]	leftOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b) Left Ore condition.
GenSolvablePolynomial<C>[]	leftSimplifier(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b) Left simplifier.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	leftZeroRelations(int modv, java.util.List<GenSolvablePolynomial<C>> F) Left syzygy for left Groebner base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	leftZeroRelations(java.util.List<GenSolvablePolynomial<C>> F) Left syzygy for left Groebner base.
ModuleList<C>	leftZeroRelations(ModuleList<C> M) Left syzygy for left module Groebner base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	leftZeroRelationsArbitrary(int modv, java.util.List<GenSolvablePolynomial<C>> F) Left syzygy module from arbitrary base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	leftZeroRelationsArbitrary(java.util.List<GenSolvablePolynomial<C>> F) Left syzygy module from arbitrary base.
ModuleList<C>	leftZeroRelationsArbitrary(ModuleList<C> M) Left syzygy for arbitrary left module base.
java.util.List<edu.jas.gbmod.SolvResPart<C>>	resolution(ModuleList<C> M) Resolution of a module.
java.util.List	resolution(PolynomialList<C> F) Resolution of a polynomial list.
java.util.List<edu.jas.gbmod.SolvResPart<C>>	resolutionArbitrary(ModuleList<C> M) Resolution of a module.
java.util.List	resolutionArbitrary(PolynomialList<C> F) Resolution of a polynomial list.
GenSolvablePolynomial<C>[]	rightOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b) Right Ore condition.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	rightZeroRelationsArbitrary(int modv, java.util.List<GenSolvablePolynomial<C>> F) Right syzygy module from arbitrary base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	rightZeroRelationsArbitrary(java.util.List<GenSolvablePolynomial<C>> F) Right syzygy module from arbitrary base.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

sred

public final SolvableReduction<C extends RingElem<C>> sred

Solvable reduction engine.

Constructor Detail

SolvableSyzygyAbstract

public SolvableSyzygyAbstract()

Constructor.

Method Detail

leftZeroRelations

public java.util.List<java.util.List<GenSolvablePolynomial<C>>> leftZeroRelations(java.util.List<GenSolvablePolynomial<C>> F)

Left syzygy for left Groebner base.

Specified by:

leftZeroRelations in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

F - a Groebner base.

Returns:

leftSyz(F), a basis for the left module of syzygies for F.

leftZeroRelations

public java.util.List<java.util.List<GenSolvablePolynomial<C>>> leftZeroRelations(int modv,
                                                                                  java.util.List<GenSolvablePolynomial<C>> F)

Left syzygy for left Groebner base.

Specified by:

leftZeroRelations in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

modv - number of module variables.

F - a Groebner base.

Returns:

leftSyz(F), a basis for the left module of syzygies for F.

leftZeroRelations

public ModuleList<C> leftZeroRelations(ModuleList<C> M)

Left syzygy for left module Groebner base.

Specified by:

leftZeroRelations in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

M - a Groebner base.

Returns:

leftSyz(M), a basis for the left module of syzygies for M.

isLeftZeroRelation

public boolean isLeftZeroRelation(java.util.List<java.util.List<GenSolvablePolynomial<C>>> Z,
                                  java.util.List<GenSolvablePolynomial<C>> F)

Test if left syzygy.

Specified by:

isLeftZeroRelation in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

Z - list of sysygies.

F - a polynomial list.

Returns:

true, if Z is a list of left syzygies for F, else false.

isRightZeroRelation

public boolean isRightZeroRelation(java.util.List<java.util.List<GenSolvablePolynomial<C>>> Z,
                                   java.util.List<GenSolvablePolynomial<C>> F)

Test if right syzygy.

Specified by:

isRightZeroRelation in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

Z - list of sysygies.

F - a polynomial list.

Returns:

true, if Z is a list of right syzygies for F, else false.

isLeftZeroRelation

public boolean isLeftZeroRelation(ModuleList<C> Z,
                                  ModuleList<C> F)

Test if left sysygy of modules

Specified by:

isLeftZeroRelation in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

Z - list of sysygies.

F - a module list.

Returns:

true, if Z is a list of left syzygies for F, else false.

isRightZeroRelation

public boolean isRightZeroRelation(ModuleList<C> Z,
                                   ModuleList<C> F)

Test if right sysygy of modules

Specified by:

isRightZeroRelation in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

Z - list of sysygies.

F - a module list.

Returns:

true, if Z is a list of right syzygies for F, else false.

resolution

public java.util.List<edu.jas.gbmod.SolvResPart<C>> resolution(ModuleList<C> M)

Resolution of a module. Only with direct GBs.

Specified by:

resolution in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

M - a module list of a Groebner basis.

Returns:

a resolution of M.

resolution

public java.util.List resolution(PolynomialList<C> F)

Resolution of a polynomial list. Only with direct GBs.

Specified by:

resolution in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

F - a polynomial list of a Groebner basis.

Returns:

a resolution of F.

resolutionArbitrary

public java.util.List<edu.jas.gbmod.SolvResPart<C>> resolutionArbitrary(ModuleList<C> M)

Resolution of a module.

Specified by:

resolutionArbitrary in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

M - a module list of an arbitrary basis.

Returns:

a resolution of M.

resolutionArbitrary

public java.util.List resolutionArbitrary(PolynomialList<C> F)

Resolution of a polynomial list.

Specified by:

resolutionArbitrary in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

F - a polynomial list of an arbitrary basis.

Returns:

a resolution of F.

leftZeroRelationsArbitrary

public java.util.List<java.util.List<GenSolvablePolynomial<C>>> leftZeroRelationsArbitrary(java.util.List<GenSolvablePolynomial<C>> F)

Left syzygy module from arbitrary base.

Specified by:

leftZeroRelationsArbitrary in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

F - a solvable polynomial list.

Returns:

syz(F), a basis for the module of left syzygies for F.

leftZeroRelationsArbitrary

public java.util.List<java.util.List<GenSolvablePolynomial<C>>> leftZeroRelationsArbitrary(int modv,
                                                                                           java.util.List<GenSolvablePolynomial<C>> F)

Left syzygy module from arbitrary base.

Specified by:

leftZeroRelationsArbitrary in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

modv - number of module variables.

F - a solvable polynomial list.

Returns:

syz(F), a basis for the module of left syzygies for F.

leftZeroRelationsArbitrary

public ModuleList<C> leftZeroRelationsArbitrary(ModuleList<C> M)

Left syzygy for arbitrary left module base.

Specified by:

leftZeroRelationsArbitrary in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

M - an arbitrary base.

Returns:

leftSyz(M), a basis for the left module of syzygies for M.

rightZeroRelationsArbitrary

public java.util.List<java.util.List<GenSolvablePolynomial<C>>> rightZeroRelationsArbitrary(java.util.List<GenSolvablePolynomial<C>> F)

Right syzygy module from arbitrary base.

Specified by:

rightZeroRelationsArbitrary in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

F - a solvable polynomial list.

Returns:

syz(F), a basis for the module of right syzygies for F.

rightZeroRelationsArbitrary

public java.util.List<java.util.List<GenSolvablePolynomial<C>>> rightZeroRelationsArbitrary(int modv,
                                                                                            java.util.List<GenSolvablePolynomial<C>> F)

Right syzygy module from arbitrary base.

Specified by:

rightZeroRelationsArbitrary in interface SolvableSyzygy<C extends RingElem<C>>

Parameters:

modv - number of module variables.

F - a solvable polynomial list.

Returns:

syz(F), a basis for the module of right syzygies for F.

isLeftOreCond

public boolean isLeftOreCond(GenSolvablePolynomial<C> a,
                             GenSolvablePolynomial<C> b,
                             GenSolvablePolynomial<C>[] oc)

Test left Ore condition.

Parameters:

a - solvable polynomial

b - solvable polynomial

oc - = [p,q] two solvable polynomials

Returns:

true if p*a = q*b, else false

isRightOreCond

public boolean isRightOreCond(GenSolvablePolynomial<C> a,
                              GenSolvablePolynomial<C> b,
                              GenSolvablePolynomial<C>[] oc)

Test right Ore condition.

Parameters:

a - solvable polynomial

b - solvable polynomial

oc - = [p,q] two solvable polynomials

Returns:

true if a*p = b*q, else false

leftOreCond

public GenSolvablePolynomial<C>[] leftOreCond(GenSolvablePolynomial<C> a,
                                              GenSolvablePolynomial<C> b)

Left Ore condition. Generators for the left Ore condition of two solvable polynomials.

Parameters:

a - solvable polynomial

b - solvable polynomial

Returns:

[p,q] with p*a = q*b

rightOreCond

public GenSolvablePolynomial<C>[] rightOreCond(GenSolvablePolynomial<C> a,
                                               GenSolvablePolynomial<C> b)

Right Ore condition. Generators for the right Ore condition of two solvable polynomials.

Parameters:

a - solvable polynomial

b - solvable polynomial

Returns:

[p,q] with a*p = b*q

leftSimplifier

public GenSolvablePolynomial<C>[] leftSimplifier(GenSolvablePolynomial<C> a,
                                                 GenSolvablePolynomial<C> b)

Left simplifier. Method of Apel & Lassner (1987).

Parameters:

a - solvable polynomial

b - solvable polynomial

Returns:

[p,q] with a/b = p/q and q is minimal and monic

compare

public int compare(GenSolvablePolynomial<C> num,
                   GenSolvablePolynomial<C> den,
                   GenSolvablePolynomial<C> n,
                   GenSolvablePolynomial<C> d)

Comparison like SolvableLocal or SolvableQuotient.

Parameters:

num - SolvablePolynomial.

den - SolvablePolynomial.

n - SolvablePolynomial.

d - SolvablePolynomial.

Returns:

sign((num/den)-(n/d)).