Package edu.jas.gb

Class ReductionAbstract<C extends RingElem<C>>

java.lang.Object

edu.jas.gb.ReductionAbstract<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

Reduction<C>, java.io.Serializable

Direct Known Subclasses:

DReductionSeq, PseudoReductionPar, PseudoReductionSeq, ReductionPar, ReductionSeq, RReductionSeq

public abstract class ReductionAbstract<C extends RingElem<C>>
extends java.lang.Object
implements Reduction<C>

Polynomial Reduction abstract class. Implements common S-Polynomial, normalform, criterion 4 module criterion and irreducible set.

Author:

Heinz Kredel

See Also:

Serialized Form

Constructor Summary

Constructors

Constructor	Description
ReductionAbstract()	Constructor.

Method Summary

Modifier and Type	Method	Description
boolean	criterion4(ExpVector ei, ExpVector ej, ExpVector e)	GB criterium 4.
boolean	criterion4(GenPolynomial<C> A, GenPolynomial<C> B)	GB criterium 4.
boolean	criterion4(GenPolynomial<C> A, GenPolynomial<C> B, ExpVector e)	GB criterium 4.
java.util.List<GenPolynomial<C>>	irreducibleSet(java.util.List<GenPolynomial<C>> Pp)	Irreducible set.
boolean	isNormalform(java.util.List<GenPolynomial<C>> Pp)	Is in Normalform.
boolean	isNormalform(java.util.List<GenPolynomial<C>> Pp, GenPolynomial<C> Ap)	Is in Normalform.
boolean	isReducible(java.util.List<GenPolynomial<C>> Pp, GenPolynomial<C> Ap)	Is reducible.
boolean	isReductionNF(java.util.List<GenPolynomial<C>> row, java.util.List<GenPolynomial<C>> Pp, GenPolynomial<C> Ap, GenPolynomial<C> Np)	Is reduction of normal form.
boolean	isTopReducible(java.util.List<GenPolynomial<C>> P, GenPolynomial<C> A)	Is top reducible.
boolean	moduleCriterion(int modv, ExpVector ei, ExpVector ej)	Module criterium.
boolean	moduleCriterion(int modv, GenPolynomial<C> A, GenPolynomial<C> B)	Module criterium.
ModuleList<C>	normalform(ModuleList<C> Pp, ModuleList<C> Ap)	Module normalform set.
ModuleList<C>	normalform(ModuleList<C> Pp, ModuleList<C> Ap, boolean top)	Module normalform set.
java.util.List<GenPolynomial<C>>	normalform(java.util.List<GenPolynomial<C>> Pp, java.util.List<GenPolynomial<C>> Ap)	Normalform Set.
GenPolynomial<C>	normalformMarked(java.util.List<Monomial<C>> Mp, java.util.List<GenPolynomial<C>> Pp, GenPolynomial<C> Ap)	Normalform with respect to marked head terms.
GenPolynomial<C>	SPolynomial(GenPolynomial<C> A, GenPolynomial<C> B)	S-Polynomial.
GenPolynomial<C>	SPolynomial(java.util.List<GenPolynomial<C>> S, int i, GenPolynomial<C> A, int j, GenPolynomial<C> B)	S-Polynomial with recording.

Methods inherited from class java.lang.Object

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

Methods inherited from interface edu.jas.gb.Reduction

normalform, normalform

Constructor Detail

ReductionAbstract

public ReductionAbstract()

Constructor.

Method Detail

SPolynomial

public GenPolynomial<C> SPolynomial(GenPolynomial<C> A,
                                    GenPolynomial<C> B)

S-Polynomial.

Specified by:

SPolynomial in interface Reduction<C extends RingElem<C>>

Parameters:

A - polynomial.

B - polynomial.

Returns:

spol(A,B) the S-polynomial of A and B.

SPolynomial

public GenPolynomial<C> SPolynomial(java.util.List<GenPolynomial<C>> S,
                                    int i,
                                    GenPolynomial<C> A,
                                    int j,
                                    GenPolynomial<C> B)

S-Polynomial with recording.

Specified by:

SPolynomial in interface Reduction<C extends RingElem<C>>

Parameters:

S - recording matrix, is modified. Note the negative S-polynomial is recorded as required by all applications.

i - index of Ap in basis list.

A - a polynomial.

j - index of Bp in basis list.

B - a polynomial.

Returns:

Spol(A, B), the S-Polynomial for A and B.

moduleCriterion

public boolean moduleCriterion(int modv,
                               GenPolynomial<C> A,
                               GenPolynomial<C> B)

Module criterium.

Specified by:

moduleCriterion in interface Reduction<C extends RingElem<C>>

Parameters:

modv - number of module variables.

A - polynomial.

B - polynomial.

Returns:

true if the module S-polynomial(i,j) is required.

moduleCriterion

public boolean moduleCriterion(int modv,
                               ExpVector ei,
                               ExpVector ej)

Module criterium.

Specified by:

moduleCriterion in interface Reduction<C extends RingElem<C>>

Parameters:

modv - number of module variables.

ei - ExpVector.

ej - ExpVector.

Returns:

true if the module S-polynomial(i,j) is required.

criterion4

public boolean criterion4(GenPolynomial<C> A,
                          GenPolynomial<C> B,
                          ExpVector e)

GB criterium 4. Use only for commutative polynomial rings.

Specified by:

criterion4 in interface Reduction<C extends RingElem<C>>

Parameters:

A - polynomial.

B - polynomial.

e - = lcm(ht(A),ht(B))

Returns:

true if the S-polynomial(i,j) is required, else false.

criterion4

public boolean criterion4(ExpVector ei,
                          ExpVector ej,
                          ExpVector e)

GB criterium 4. Use only for commutative polynomial rings.

Specified by:

criterion4 in interface Reduction<C extends RingElem<C>>

Parameters:

ei - exponent vector.

ej - exponent vector.

e - = lcm(ei,ej)

Returns:

true if the S-polynomial(i,j) is required, else false.

criterion4

public boolean criterion4(GenPolynomial<C> A,
                          GenPolynomial<C> B)

GB criterium 4.

Specified by:

criterion4 in interface Reduction<C extends RingElem<C>>

Parameters:

A - polynomial.

B - polynomial.

Returns:

true if the S-polynomial(i,j) is required, else false.

normalformMarked

public GenPolynomial<C> normalformMarked(java.util.List<Monomial<C>> Mp,
                                         java.util.List<GenPolynomial<C>> Pp,
                                         GenPolynomial<C> Ap)

Normalform with respect to marked head terms.

Parameters:

Mp - leading monomial list.

Pp - polynomial list.

Ap - polynomial.

Returns:

nf(Ap) with respect to Mp+Pp.

normalform

public java.util.List<GenPolynomial<C>> normalform(java.util.List<GenPolynomial<C>> Pp,
                                                   java.util.List<GenPolynomial<C>> Ap)

Normalform Set.

Specified by:

normalform in interface Reduction<C extends RingElem<C>>

Parameters:

Ap - polynomial list.

Pp - polynomial list.

Returns:

list of nf(a) with respect to Pp for all a in Ap.

normalform

public ModuleList<C> normalform(ModuleList<C> Pp,
                                ModuleList<C> Ap)

Module normalform set.

Parameters:

Ap - module list.

Pp - module list.

Returns:

list of nf(a) with respect to Pp for all a in Ap.

normalform

public ModuleList<C> normalform(ModuleList<C> Pp,
                                ModuleList<C> Ap,
                                boolean top)

Module normalform set.

Parameters:

Ap - module list.

Pp - module list.

top - true for TOP term order, false for POT term order.

Returns:

list of nf(a) with respect to Pp for all a in Ap.

isTopReducible

public boolean isTopReducible(java.util.List<GenPolynomial<C>> P,
                              GenPolynomial<C> A)

Is top reducible.

Specified by:

isTopReducible in interface Reduction<C extends RingElem<C>>

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is top reducible with respect to P.

isReducible

public boolean isReducible(java.util.List<GenPolynomial<C>> Pp,
                           GenPolynomial<C> Ap)

Is reducible.

Specified by:

isReducible in interface Reduction<C extends RingElem<C>>

Parameters:

Ap - polynomial.

Pp - polynomial list.

Returns:

true if Ap is reducible with respect to Pp.

isNormalform

public boolean isNormalform(java.util.List<GenPolynomial<C>> Pp,
                            GenPolynomial<C> Ap)

Is in Normalform.

Specified by:

isNormalform in interface Reduction<C extends RingElem<C>>

Parameters:

Ap - polynomial.

Pp - polynomial list.

Returns:

true if Ap is in normalform with respect to Pp.

isNormalform

public boolean isNormalform(java.util.List<GenPolynomial<C>> Pp)

Is in Normalform.

Specified by:

isNormalform in interface Reduction<C extends RingElem<C>>

Parameters:

Pp - polynomial list.

Returns:

true if each Ap in Pp is in normalform with respect to Pp\{Ap}.

irreducibleSet

public java.util.List<GenPolynomial<C>> irreducibleSet(java.util.List<GenPolynomial<C>> Pp)

Irreducible set.

Specified by:

irreducibleSet in interface Reduction<C extends RingElem<C>>

Parameters:

Pp - polynomial list.

Returns:

a list P of monic polynomials which are in normalform wrt. P and with ideal(Pp) = ideal(P).

isReductionNF

public boolean isReductionNF(java.util.List<GenPolynomial<C>> row,
                             java.util.List<GenPolynomial<C>> Pp,
                             GenPolynomial<C> Ap,
                             GenPolynomial<C> Np)

Is reduction of normal form.

Specified by:

isReductionNF in interface Reduction<C extends RingElem<C>>

Parameters:

row - recording matrix.

Pp - a polynomial list for reduction.

Ap - a polynomial.

Np - nf(Pp,Ap), a normal form of Ap wrt. Pp.

Returns:

true, if Np + sum( row[i]*Pp[i] ) == Ap, else false.