edu.jas.gb
Interface Reduction<C extends RingElem<C>>

Type Parameters:

C - coefficient type

All Superinterfaces:

java.io.Serializable

All Known Subinterfaces:

DReduction<C>, EReduction<C>, PseudoReduction<C>, RPseudoReduction<C>, RReduction<C>

All Known Implementing Classes:

DReductionSeq, EReductionSeq, PseudoReductionSeq, ReductionAbstract, ReductionPar, ReductionSeq, RPseudoReductionSeq, RReductionSeq

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

Polynomial Reduction interface. Defines S-Polynomial, normalform, criterion 4, module criterion and irreducible set.

Author:

Heinz Kredel

Method Summary

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>> P, GenPolynomial<C> A) Is in Normalform.
boolean	isReducible(java.util.List<GenPolynomial<C>> P, GenPolynomial<C> A) 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.
GenPolynomial<C>	normalform(java.util.List<GenPolynomial<C>> P, GenPolynomial<C> A) Normalform.
java.util.List<GenPolynomial<C>>	normalform(java.util.List<GenPolynomial<C>> Pp, java.util.List<GenPolynomial<C>> Ap) Normalform Set.
GenPolynomial<C>	normalform(java.util.List<GenPolynomial<C>> row, java.util.List<GenPolynomial<C>> Pp, GenPolynomial<C> Ap) Normalform with recording.
GenPolynomial<C>	SPolynomial(GenPolynomial<C> Ap, GenPolynomial<C> Bp) S-Polynomial.
GenPolynomial<C>	SPolynomial(java.util.List<GenPolynomial<C>> S, int i, GenPolynomial<C> Ap, int j, GenPolynomial<C> Bp) S-Polynomial with recording.

Method Detail

SPolynomial

GenPolynomial<C> SPolynomial(GenPolynomial<C> Ap,
                             GenPolynomial<C> Bp)

S-Polynomial.

Parameters:

Ap - polynomial.

Bp - polynomial.

Returns:

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

SPolynomial

GenPolynomial<C> SPolynomial(java.util.List<GenPolynomial<C>> S,
                             int i,
                             GenPolynomial<C> Ap,
                             int j,
                             GenPolynomial<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:

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

moduleCriterion

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

Module criterium.

Parameters:

modv - number of module variables.

A - polynomial.

B - polynomial.

Returns:

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

moduleCriterion

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

Module criterium.

Parameters:

modv - number of module variables.

ei - ExpVector.

ej - ExpVector.

Returns:

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

criterion4

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

GB criterium 4. Use only for commutative polynomial rings.

Parameters:

A - polynomial.

B - polynomial.

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

Returns:

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

criterion4

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

GB criterium 4. Use only for commutative polynomial rings.

Parameters:

A - polynomial.

B - polynomial.

Returns:

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

isTopReducible

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

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

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is top reducible with respect to P.

isReducible

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

Is reducible.

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is reducible with respect to P.

isNormalform

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

Is in Normalform.

Parameters:

A - polynomial.

P - polynomial list.

Returns:

true if A is in normalform with respect to P.

isNormalform

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

Is in Normalform.

Parameters:

Pp - polynomial list.

Returns:

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

normalform

GenPolynomial<C> normalform(java.util.List<GenPolynomial<C>> P,
                            GenPolynomial<C> A)

Normalform.

Parameters:

A - polynomial.

P - polynomial list.

Returns:

nf(A) with respect to P.

normalform

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

Normalform Set.

Parameters:

Ap - polynomial list.

Pp - polynomial list.

Returns:

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

normalform

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

Normalform with recording.

Parameters:

row - recording matrix, is modified.

Pp - a polynomial list for reduction.

Ap - a polynomial.

Returns:

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

irreducibleSet

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

Irreducible set.

Parameters:

Pp - polynomial list.

Returns:

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

isReductionNF

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.

Parameters:

row - recording matrix, is modified.

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.