Class SolvableSyzygyAbstract<C extends GcdRingElem<C>>

java.lang.Object

edu.jas.gbufd.SolvableSyzygyAbstract<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

SolvableSyzygy<C>, Serializable

Direct Known Subclasses:

SolvableSyzygySeq

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

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

See Also:

• Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
protected BasicLinAlg<GenPolynomial<C>>	blas	Linear algebra engine.
private static final boolean	debug
private static final org.apache.logging.log4j.Logger	logger
protected Reduction<C>	red	Reduction engine.
final 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	isLeftOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b, GenSolvablePolynomial<C> p, GenSolvablePolynomial<C> q)	Is left Ore condition.
boolean	isLeftZeroRelation(ModuleList<C> Z, ModuleList<C> F)	Test if left sysygy of modules
boolean	isLeftZeroRelation(List<List<GenSolvablePolynomial<C>>> Z, List<GenSolvablePolynomial<C>> F)	Test if left syzygy.
boolean	isRightOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b, GenSolvablePolynomial<C>[] oc)	Test right Ore condition.
boolean	isRightOreCond(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b, GenSolvablePolynomial<C> p, GenSolvablePolynomial<C> q)	Is right Ore condition.
boolean	isRightZeroRelation(ModuleList<C> Z, ModuleList<C> F)	Test if right sysygy of modules
boolean	isRightZeroRelation(List<List<GenSolvablePolynomial<C>>> Z, List<GenSolvablePolynomial<C>> F)	Test if right syzygy.
abstract GenSolvablePolynomial<C>[]	leftSimplifier(GenSolvablePolynomial<C> a, GenSolvablePolynomial<C> b)	Left simplifier.
List<List<GenSolvablePolynomial<C>>>	leftZeroRelations(int modv, List<GenSolvablePolynomial<C>> F)	Left syzygy for left Groebner base.
ModuleList<C>	leftZeroRelations(ModuleList<C> M)	Left syzygy for left module Groebner base.
List<List<GenSolvablePolynomial<C>>>	leftZeroRelations(List<GenSolvablePolynomial<C>> F)	Left syzygy for left Groebner base.
ModuleList<C>	leftZeroRelationsArbitrary(ModuleList<C> M)	Left syzygy for arbitrary left module base.
List<List<GenSolvablePolynomial<C>>>	leftZeroRelationsArbitrary(List<GenSolvablePolynomial<C>> F)	Left syzygy module from arbitrary base.
List<List<GenSolvablePolynomial<C>>>	rightZeroRelations(int modv, List<GenSolvablePolynomial<C>> F)	Right syzygy module from Groebner base.
ModuleList<C>	rightZeroRelations(ModuleList<C> M)	Right syzygy for right module Groebner base.
List<List<GenSolvablePolynomial<C>>>	rightZeroRelations(List<GenSolvablePolynomial<C>> F)	Right syzygy module from Groebner base.
List<List<GenSolvablePolynomial<C>>>	rightZeroRelationsArbitrary(int modv, List<GenSolvablePolynomial<C>> F)	Right syzygy module from arbitrary base.
ModuleList<C>	rightZeroRelationsArbitrary(ModuleList<C> M)	Right syzygy for arbitrary base.
List<List<GenSolvablePolynomial<C>>>	rightZeroRelationsArbitrary(List<GenSolvablePolynomial<C>> F)	Right syzygy module from arbitrary base.

Methods inherited from class Object

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

Methods inherited from interface SolvableSyzygy

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

Field Details

logger

private static final org.apache.logging.log4j.Logger logger

debug

private static final boolean debug

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 Details

SolvableSyzygyAbstract

public SolvableSyzygyAbstract()

Constructor.

Method Details

leftZeroRelations

public List<List<GenSolvablePolynomial<C>>> leftZeroRelations(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 List<List<GenSolvablePolynomial<C>>> leftZeroRelations(int modv,
 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 List<List<GenSolvablePolynomial<C>>> rightZeroRelations(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 List<List<GenSolvablePolynomial<C>>> rightZeroRelations(int modv,
 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(List<List<GenSolvablePolynomial<C>>> Z,
 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(List<List<GenSolvablePolynomial<C>>> Z,
 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 List<List<GenSolvablePolynomial<C>>> leftZeroRelationsArbitrary(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 List<List<GenSolvablePolynomial<C>>> rightZeroRelationsArbitrary(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 List<List<GenSolvablePolynomial<C>>> rightZeroRelationsArbitrary(int modv,
 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 arbitrary 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

isLeftOreCond

public boolean isLeftOreCond(GenSolvablePolynomial<C> a,
 GenSolvablePolynomial<C> b,
 GenSolvablePolynomial<C> p,
 GenSolvablePolynomial<C> q)

Is left Ore condition. Test left Ore condition of two solvable polynomials.

Parameters:

a - solvable polynomial

b - solvable polynomial

p - solvable polynomial

q - solvable polynomial

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

isRightOreCond

public boolean isRightOreCond(GenSolvablePolynomial<C> a,
 GenSolvablePolynomial<C> b,
 GenSolvablePolynomial<C> p,
 GenSolvablePolynomial<C> q)

Is right Ore condition. Test right Ore condition of two solvable polynomials.

Parameters:

a - solvable polynomial

b - solvable polynomial

p - solvable polynomial

q - solvable polynomial

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)).