Package edu.jas.gb

Class SigReductionSeq<C extends RingElem<C>>

java.lang.Object

edu.jas.gb.SigReductionSeq<C>

Type Parameters:

C - coefficient type

All Implemented Interfaces:

SigReduction<C>, java.io.Serializable

public class SigReductionSeq<C extends RingElem<C>>
extends java.lang.Object
implements SigReduction<C>

Polynomial SigReduction class. Implements common S-Polynomial, normalform with respect to signatures.

Author:

Heinz Kredel

See Also:

Serialized Form

Constructor Summary

Constructors

Constructor	Description
SigReductionSeq()	Constructor.

Method Summary

Modifier and Type	Method	Description
boolean	isSigNormalform(java.util.List<SigPoly<C>> F, java.util.List<SigPoly<C>> G, SigPoly<C> A)	Is in top normalform.
boolean	isSigReducible(java.util.List<SigPoly<C>> F, java.util.List<SigPoly<C>> G, SigPoly<C> A)	Is top reducible.
boolean	isSigRedundant(java.util.List<SigPoly<C>> G, SigPoly<C> A)	Is sigma redundant.
boolean	isSigRedundantAlt(java.util.List<SigPoly<C>> G, SigPoly<C> A)	Is sigma redundant, alternative algorithm.
java.util.List<SigPair<C>>[]	minDegSubset(java.util.List<SigPair<C>> F)	Select signature polynomials of minimal degree and non minimal degree.
long	minimalSigDegree(java.util.List<SigPair<C>> F)	Minimal degree of signatures.
java.util.List<GenPolynomial<C>>	polys(java.util.List<SigPoly<C>> F)	Select polynomials.
java.util.List<GenPolynomial<C>>	sigmas(java.util.List<SigPair<C>> F)	Select signatures.
SigPoly<C>	sigNormalform(java.util.List<GenPolynomial<C>> F, java.util.List<SigPoly<C>> G, SigPoly<C> A)	Top normalform.
SigPoly<C>	sigSemiNormalform(java.util.List<GenPolynomial<C>> F, java.util.List<SigPoly<C>> G, SigPoly<C> A)	Top semi-complete normalform.
java.util.List<SigPair<C>>	sortSigma(java.util.List<SigPair<C>> F)	Sort signature polynomials according to the degree its signatures.
GenPolynomial<C>	SPolynomial(SigPoly<C> A, SigPoly<C> B)	S-Polynomial.
ExpVector[]	SPolynomialExpVectorFactors(SigPoly<C> A, SigPoly<C> B)	S-Polynomial factors.
GenPolynomial<C>[]	SPolynomialFactors(SigPoly<C> A, SigPoly<C> B)	S-Polynomial polynomial factors.
GenPolynomial<C>	SPolynomialHalf(SigPoly<C> A, SigPoly<C> B)	S-Polynomial half.

Methods inherited from class java.lang.Object

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

Constructor Detail

SigReductionSeq

public SigReductionSeq()

Constructor.

Method Detail

SPolynomial

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

S-Polynomial.

Specified by:

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

Parameters:

A - polynomial.

B - polynomial.

Returns:

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

SPolynomialExpVectorFactors

public ExpVector[] SPolynomialExpVectorFactors(SigPoly<C> A,
                                               SigPoly<C> B)

S-Polynomial factors.

Parameters:

A - monic polynomial.

B - monic polynomial.

Returns:

exponent vectors [e,f] such that spol(A,B) = e*a - f*B.

SPolynomialHalf

public GenPolynomial<C> SPolynomialHalf(SigPoly<C> A,
                                        SigPoly<C> B)

S-Polynomial half.

Parameters:

A - monic polynomial.

B - monic polynomial.

Returns:

e*A "half" of an S-polynomial such that spol(A,B) = e*A - f*B.

SPolynomialFactors

public GenPolynomial<C>[] SPolynomialFactors(SigPoly<C> A,
                                             SigPoly<C> B)

S-Polynomial polynomial factors.

Parameters:

A - monic polynomial.

B - monic polynomial.

Returns:

polynomials [e,f] such that spol(A,B) = e*a - f*B.

isSigReducible

public boolean isSigReducible(java.util.List<SigPoly<C>> F,
                              java.util.List<SigPoly<C>> G,
                              SigPoly<C> A)

Is top reducible. Condition is lt(B) | lt(A) for some B in F or G.

Specified by:

isSigReducible in interface SigReduction<C extends RingElem<C>>

Parameters:

A - polynomial.

F - polynomial list.

G - polynomial list.

Returns:

true if A is top reducible with respect to F and G.

isSigNormalform

public boolean isSigNormalform(java.util.List<SigPoly<C>> F,
                               java.util.List<SigPoly<C>> G,
                               SigPoly<C> A)

Is in top normalform.

Specified by:

isSigNormalform in interface SigReduction<C extends RingElem<C>>

Parameters:

A - polynomial.

F - polynomial list.

G - polynomial list.

Returns:

true if A is in top normalform with respect to F and G.

isSigRedundant

public boolean isSigRedundant(java.util.List<SigPoly<C>> G,
                              SigPoly<C> A)

Is sigma redundant.

Parameters:

A - polynomial.

G - polynomial list.

Returns:

true if A is sigma redundant with respect to G.

isSigRedundantAlt

public boolean isSigRedundantAlt(java.util.List<SigPoly<C>> G,
                                 SigPoly<C> A)

Is sigma redundant, alternative algorithm.

Parameters:

A - polynomial.

G - polynomial list.

Returns:

true if A is sigma redundant per alternative algorithm with respect to G.

sigNormalform

public SigPoly<C> sigNormalform(java.util.List<GenPolynomial<C>> F,
                                java.util.List<SigPoly<C>> G,
                                SigPoly<C> A)

Top normalform.

Specified by:

sigNormalform in interface SigReduction<C extends RingElem<C>>

Parameters:

A - polynomial.

F - polynomial list.

G - polynomial list.

Returns:

nf(A) with respect to F and G.

sigSemiNormalform

public SigPoly<C> sigSemiNormalform(java.util.List<GenPolynomial<C>> F,
                                    java.util.List<SigPoly<C>> G,
                                    SigPoly<C> A)

Top semi-complete normalform.

Parameters:

A - polynomial.

F - polynomial list.

G - polynomial list.

Returns:

nf(A) with respect to F and G.

polys

public java.util.List<GenPolynomial<C>> polys(java.util.List<SigPoly<C>> F)

Select polynomials.

Parameters:

F - list of signature polynomials.

Returns:

the polynomials in F.

sigmas

public java.util.List<GenPolynomial<C>> sigmas(java.util.List<SigPair<C>> F)

Select signatures.

Parameters:

F - list of signature polynomials.

Returns:

the signatures in F.

minimalSigDegree

public long minimalSigDegree(java.util.List<SigPair<C>> F)

Minimal degree of signatures.

Parameters:

F - list of signature polynomials.

Returns:

the minimal degree of the signatures in F.

minDegSubset

public java.util.List<SigPair<C>>[] minDegSubset(java.util.List<SigPair<C>> F)

Select signature polynomials of minimal degree and non minimal degree.

Parameters:

F - list of signature polynomials.

Returns:

[m,p] where m is the list of signature polynomials of F of minimal degree and p contains the rest of the signature polynomials with non minimal degree.

sortSigma

public java.util.List<SigPair<C>> sortSigma(java.util.List<SigPair<C>> F)

Sort signature polynomials according to the degree its signatures.

Parameters:

F - list of signature polynomials.

Returns:

list of signature polynomials sorted by degree of sigma.