Package edu.jas.gbufd

Class SolvableSyzygyAbstract<C extends GcdRingElem<C>>

java.lang.Object

edu.jas.gbufd.SolvableSyzygyAbstract<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

SolvableSyzygy<C>, java.io.Serializable

Direct Known Subclasses:

SolvableSyzygySeq

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

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

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
protected BasicLinAlg<GenPolynomial<C>>	blas	Linear algebra engine.
protected Reduction<C>	red	Reduction engine.
SolvableReduction<C>	sred	Solvable reduction engine.

Constructor Summary

Constructors

Constructor	Description
SolvableSyzygyAbstract()	Constructor.

Method Summary

Modifier and Type	Method	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(ModuleList<C> Z, ModuleList<C> F)	Test if left sysygy of modules
boolean	isLeftZeroRelation(java.util.List<java.util.List<GenSolvablePolynomial<C>>> Z, java.util.List<GenSolvablePolynomial<C>> F)	Test if left syzygy.
boolean	isRightOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b, GenSolvablePolynomial<C>[] oc)	Test right Ore condition.
boolean	isRightZeroRelation(ModuleList<C> Z, ModuleList<C> F)	Test if right sysygy of modules
boolean	isRightZeroRelation(java.util.List<java.util.List<GenSolvablePolynomial<C>>> Z, java.util.List<GenSolvablePolynomial<C>> F)	Test if right syzygy.
abstract 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.
ModuleList<C>	leftZeroRelations(ModuleList<C> M)	Left syzygy for left module 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>	leftZeroRelationsArbitrary(ModuleList<C> M)	Left syzygy for arbitrary left module base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	leftZeroRelationsArbitrary(java.util.List<GenSolvablePolynomial<C>> F)	Left syzygy module from arbitrary base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	rightZeroRelations(int modv, java.util.List<GenSolvablePolynomial<C>> F)	Right syzygy module from Groebner base.
ModuleList<C>	rightZeroRelations(ModuleList<C> M)	Right syzygy for right module Groebner base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	rightZeroRelations(java.util.List<GenSolvablePolynomial<C>> F)	Right syzygy module from Groebner base.
java.util.List<java.util.List<GenSolvablePolynomial<C>>>	rightZeroRelationsArbitrary(int modv, java.util.List<GenSolvablePolynomial<C>> F)	Right syzygy module from arbitrary base.
ModuleList<C>	rightZeroRelationsArbitrary(ModuleList<C> M)	Right syzygy for 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

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

Methods inherited from interface edu.jas.gbufd.SolvableSyzygy

leftOreCond, leftZeroRelationsArbitrary, resolution, resolution, resolutionArbitrary, resolutionArbitrary, rightOreCond

Field Detail

sred

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

Solvable reduction engine.

red

protected Reduction<C extends GcdRingElem<C>> red

Reduction engine.

blas

protected BasicLinAlg<GenPolynomial<C extends GcdRingElem<C>>> blas

Linear algebra 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 GcdRingElem<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 GcdRingElem<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 GcdRingElem<C>>

Parameters:

M - a Groebner base.

Returns:

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

rightZeroRelations

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

Right syzygy module from Groebner base.

Parameters:

F - a solvable polynomial list, a Groebner base.

Returns:

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

rightZeroRelations

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

Right syzygy module from Groebner base.

Parameters:

modv - number of module variables.

F - a solvable polynomial list, a Groebner base.

Returns:

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

rightZeroRelations

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

Right syzygy for right module Groebner base.

Parameters:

M - a Groebner base.

Returns:

rightSyz(M), a basis for the right 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 GcdRingElem<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 GcdRingElem<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 GcdRingElem<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 GcdRingElem<C>>

Parameters:

Z - list of sysygies.

F - a module list.

Returns:

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

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 GcdRingElem<C>>

Parameters:

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 GcdRingElem<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 GcdRingElem<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 GcdRingElem<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.

rightZeroRelationsArbitrary

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

Right syzygy for arbitrary base.

Parameters:

M - an arbitray base.

Returns:

rightSyz(M), a basis for the right module of syzygies for M.

isLeftOreCond

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

Test left Ore condition.

Specified by:

isLeftOreCond in interface SolvableSyzygy<C extends GcdRingElem<C>>

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.

Specified by:

isRightOreCond in interface SolvableSyzygy<C extends GcdRingElem<C>>

Parameters:

a - solvable polynomial

b - solvable polynomial

oc - = [p,q] two solvable polynomials

Returns:

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

leftSimplifier

public abstract 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)).