Package edu.jas.gb

Interface SolvableReduction<C extends RingElem<C>>

Type Parameters:

C - coefficient type

All Superinterfaces:

java.io.Serializable

All Known Subinterfaces:

SolvablePseudoReduction<C>

All Known Implementing Classes:

SolvablePseudoReductionSeq, SolvableReductionAbstract, SolvableReductionPar, SolvableReductionSeq

public interface SolvableReduction<C extends RingElem<C>>
extends java.io.Serializable

Solvable polynomial Reduction interface. Defines S-Polynomial, normalform and irreducible set.

Author:

Heinz Kredel

Method Summary

Modifier and Type	Method	Description
boolean	isLeftReductionNF(java.util.List<GenSolvablePolynomial<C>> row, java.util.List<GenSolvablePolynomial<C>> Pp, GenSolvablePolynomial<C> Ap, GenSolvablePolynomial<C> Np)	Is reduction of normal form.
boolean	isNormalform(java.util.List<GenSolvablePolynomial<C>> P, GenSolvablePolynomial<C> A)	Is in normalform.
boolean	isReducible(java.util.List<GenSolvablePolynomial<C>> P, GenSolvablePolynomial<C> A)	Is reducible.
boolean	isRightReductionNF(java.util.List<GenSolvablePolynomial<C>> row, java.util.List<GenSolvablePolynomial<C>> Pp, GenSolvablePolynomial<C> Ap, GenSolvablePolynomial<C> Np)	Is right reduction of normal form.
boolean	isTopReducible(java.util.List<GenSolvablePolynomial<C>> P, GenSolvablePolynomial<C> A)	Is top reducible.
java.util.List<GenSolvablePolynomial<C>>	leftIrreducibleSet(java.util.List<GenSolvablePolynomial<C>> Pp)	Left irreducible set.
GenSolvablePolynomial<C>	leftNormalform(java.util.List<GenSolvablePolynomial<C>> Pp, GenSolvablePolynomial<C> Ap)	Left Normalform.
java.util.List<GenSolvablePolynomial<C>>	leftNormalform(java.util.List<GenSolvablePolynomial<C>> Pp, java.util.List<GenSolvablePolynomial<C>> Ap)	Left Normalform Set.
GenSolvablePolynomial<C>	leftNormalform(java.util.List<GenSolvablePolynomial<C>> row, java.util.List<GenSolvablePolynomial<C>> Pp, GenSolvablePolynomial<C> Ap)	LeftNormalform with recording.
GenSolvablePolynomial<C>	leftSPolynomial(GenSolvablePolynomial<C> Ap, GenSolvablePolynomial<C> Bp)	Left S-Polynomial.
GenSolvablePolynomial<C>	leftSPolynomial(java.util.List<GenSolvablePolynomial<C>> S, int i, GenSolvablePolynomial<C> Ap, int j, GenSolvablePolynomial<C> Bp)	S-Polynomial with recording.
GenSolvablePolynomial<C>	normalform(java.util.List<GenSolvablePolynomial<C>> Pp, GenSolvablePolynomial<C> Ap)	Two-sided Normalform.
GenSolvablePolynomial<C>	rightNormalform(java.util.List<GenSolvablePolynomial<C>> Pp, GenSolvablePolynomial<C> Ap)	Right Normalform.
GenSolvablePolynomial<C>	rightNormalform(java.util.List<GenSolvablePolynomial<C>> row, java.util.List<GenSolvablePolynomial<C>> Pp, GenSolvablePolynomial<C> Ap)	RightNormalform with recording.
GenSolvablePolynomial<C>	rightSPolynomial(GenSolvablePolynomial<C> Ap, GenSolvablePolynomial<C> Bp)	Right S-Polynomial.

Method Detail

leftSPolynomial

GenSolvablePolynomial<C> leftSPolynomial(GenSolvablePolynomial<C> Ap,
                                         GenSolvablePolynomial<C> Bp)

Left S-Polynomial.

Parameters:

Ap - solvable polynomial.

Bp - solvable polynomial.

Returns:

left-spol(Ap,Bp) the left S-polynomial of Ap and Bp.

leftSPolynomial

GenSolvablePolynomial<C> leftSPolynomial(java.util.List<GenSolvablePolynomial<C>> S,
                                         int i,
                                         GenSolvablePolynomial<C> Ap,
                                         int j,
                                         GenSolvablePolynomial<C> Bp)

S-Polynomial with recording.

Parameters:

S - recording matrix, is modified.

i - index of Ap in basis list.

Ap - a polynomial.

j - index of Bp in basis list.

Bp - a polynomial.

Returns:

leftSpol(Ap, Bp), the left S-Polynomial for Ap and Bp.

leftNormalform

GenSolvablePolynomial<C> leftNormalform(java.util.List<GenSolvablePolynomial<C>> Pp,
                                        GenSolvablePolynomial<C> Ap)

Left Normalform.

Parameters:

Ap - solvable polynomial.

Pp - solvable polynomial list.

Returns:

left-nf(Ap) with respect to Pp.

leftNormalform

GenSolvablePolynomial<C> leftNormalform(java.util.List<GenSolvablePolynomial<C>> row,
                                        java.util.List<GenSolvablePolynomial<C>> Pp,
                                        GenSolvablePolynomial<C> Ap)

LeftNormalform with recording.

Parameters:

row - recording matrix, is modified.

Pp - a polynomial list for reduction.

Ap - a polynomial.

Returns:

nf(Pp,Ap), the left normal form of Ap wrt. Pp.

leftNormalform

java.util.List<GenSolvablePolynomial<C>> leftNormalform(java.util.List<GenSolvablePolynomial<C>> Pp,
                                                        java.util.List<GenSolvablePolynomial<C>> Ap)

Left Normalform Set.

Parameters:

Ap - solvable polynomial list.

Pp - solvable polynomial list.

Returns:

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

leftIrreducibleSet

java.util.List<GenSolvablePolynomial<C>> leftIrreducibleSet(java.util.List<GenSolvablePolynomial<C>> Pp)

Left irreducible set.

Parameters:

Pp - solvable polynomial list.

Returns:

a list P of solvable polynomials which are in normalform wrt. P.

isLeftReductionNF

boolean isLeftReductionNF(java.util.List<GenSolvablePolynomial<C>> row,
                          java.util.List<GenSolvablePolynomial<C>> Pp,
                          GenSolvablePolynomial<C> Ap,
                          GenSolvablePolynomial<C> Np)

Is reduction of normal form.

Parameters:

row - recording matrix, is modified.

Pp - a solvable polynomial list for reduction.

Ap - a solvable polynomial.

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

Returns:

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

rightSPolynomial

GenSolvablePolynomial<C> rightSPolynomial(GenSolvablePolynomial<C> Ap,
                                          GenSolvablePolynomial<C> Bp)

Right S-Polynomial.

Parameters:

Ap - solvable polynomial.

Bp - solvable polynomial.

Returns:

right-spol(Ap,Bp) the right S-polynomial of Ap and Bp.

rightNormalform

GenSolvablePolynomial<C> rightNormalform(java.util.List<GenSolvablePolynomial<C>> Pp,
                                         GenSolvablePolynomial<C> Ap)

Right Normalform.

Parameters:

Ap - solvable polynomial.

Pp - solvable polynomial list.

Returns:

right-nf(Ap) with respect to Pp.

rightNormalform

GenSolvablePolynomial<C> rightNormalform(java.util.List<GenSolvablePolynomial<C>> row,
                                         java.util.List<GenSolvablePolynomial<C>> Pp,
                                         GenSolvablePolynomial<C> Ap)

RightNormalform with recording.

Parameters:

row - recording matrix, is modified.

Pp - a polynomial list for reduction.

Ap - a polynomial.

Returns:

nf(Pp,Ap), the right normal form of Ap wrt. Pp.

isTopReducible

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

Is top reducible. Condition is lt(B) | lt(A) for some B in F. Is left right symmetric.

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is top reducible with respect to P.

isReducible

boolean isReducible(java.util.List<GenSolvablePolynomial<C>> P,
                    GenSolvablePolynomial<C> A)

Is reducible. Is left right symmetric.

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is reducible with respect to P.

isNormalform

boolean isNormalform(java.util.List<GenSolvablePolynomial<C>> P,
                     GenSolvablePolynomial<C> A)

Is in normalform. Is left right symmetric.

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is in normalform with respect to P.

isRightReductionNF

boolean isRightReductionNF(java.util.List<GenSolvablePolynomial<C>> row,
                           java.util.List<GenSolvablePolynomial<C>> Pp,
                           GenSolvablePolynomial<C> Ap,
                           GenSolvablePolynomial<C> Np)

Is right reduction of normal form.

Parameters:

row - recording matrix, is modified.

Pp - a solvable polynomial list for reduction.

Ap - a solvable polynomial.

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

Returns:

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

normalform

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

Two-sided Normalform.

Parameters:

Ap - solvable polynomial.

Pp - solvable polynomial list.

Returns:

two-sided-nf(Ap) with respect to Pp.