Package edu.jas.gb
Interface SolvableReduction<C extends RingElem<C>>
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- 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
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Method Summary
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Method Detail
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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