edu.jas.ufd

Class SquarefreeAbstract<C extends GcdRingElem<C>>

java.lang.Object

edu.jas.ufd.SquarefreeAbstract<C>

All Implemented Interfaces:

Squarefree<C>, java.io.Serializable

Direct Known Subclasses:

SquarefreeFieldChar0, SquarefreeFieldCharP, SquarefreeRingChar0

public abstract class SquarefreeAbstract<C extends GcdRingElem<C>>
extends java.lang.Object
implements Squarefree<C>

Abstract squarefree decomposition class.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
protected GreatestCommonDivisorAbstract<C>	engine GCD engine for respective base coefficients.

Constructor Summary

Constructors

Constructor and Description
SquarefreeAbstract(GreatestCommonDivisorAbstract<C> engine) Constructor.

Method Summary

Modifier and Type	Method and Description
java.util.List<java.util.List<GenPolynomial<C>>>	basePartialFraction(GenPolynomial<C> A, java.util.SortedMap<GenPolynomial<C>,java.lang.Long> D) Univariate GenPolynomial partial fraction decomposition.
abstract java.util.SortedMap<GenPolynomial<C>,java.lang.Long>	baseSquarefreeFactors(GenPolynomial<C> A) GenPolynomial polynomial squarefree factorization.
abstract GenPolynomial<C>	baseSquarefreePart(GenPolynomial<C> P) GenPolynomial polynomial greatest squarefree divisor.
java.util.List<GenPolynomial<C>>	coPrimeSquarefree(GenPolynomial<C> a, java.util.List<GenPolynomial<C>> P) GenPolynomial squarefree and co-prime list.
java.util.List<GenPolynomial<C>>	coPrimeSquarefree(java.util.List<GenPolynomial<C>> A) GenPolynomial squarefree and co-prime list.
long	factorCount(java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F) Count number of factors in a (squarefree) factorization.
boolean	isBasePartialFraction(GenPolynomial<C> A, java.util.SortedMap<GenPolynomial<C>,java.lang.Long> D, java.util.List<java.util.List<GenPolynomial<C>>> F) Test for Univariate GenPolynomial partial fraction decomposition.
boolean	isCoPrimeSquarefree(java.util.List<GenPolynomial<C>> B) Test if list of GenPolynomials is squarefree and co-prime.
boolean	isFactorization(GenPolynomial<C> P, java.util.List<GenPolynomial<C>> F) GenPolynomial is (squarefree) factorization.
boolean	isFactorization(GenPolynomial<C> P, java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F) GenPolynomial is (squarefree) factorization.
boolean	isRecursiveFactorization(GenPolynomial<GenPolynomial<C>> P, java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long> F) GenPolynomial is (squarefree) factorization.
boolean	isRecursiveSquarefree(GenPolynomial<GenPolynomial<C>> P) Recursive GenPolynomial test if is squarefree.
boolean	isSquarefree(GenPolynomial<C> P) GenPolynomial test if is squarefree.
boolean	isSquarefree(java.util.List<GenPolynomial<C>> L) GenPolynomial list test if squarefree.
java.util.SortedMap<GenPolynomial<C>,java.lang.Long>	normalizeFactorization(java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F) Normalize factorization. p'_i > 0 for i > 1 and p'_1 !
java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long>	recursiveSquarefreeFactors(GenPolynomial<GenPolynomial<C>> P) GenPolynomial recursive polynomial squarefree factorization.
GenPolynomial<GenPolynomial<C>>	recursiveSquarefreePart(GenPolynomial<GenPolynomial<C>> P) GenPolynomial recursive polynomial greatest squarefree divisor.
abstract java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long>	recursiveUnivariateSquarefreeFactors(GenPolynomial<GenPolynomial<C>> P) GenPolynomial recursive univariate polynomial squarefree factorization.
abstract GenPolynomial<GenPolynomial<C>>	recursiveUnivariateSquarefreePart(GenPolynomial<GenPolynomial<C>> P) GenPolynomial recursive polynomial greatest squarefree divisor.
abstract java.util.SortedMap<C,java.lang.Long>	squarefreeFactors(C P) Coefficients squarefree factorization.
abstract java.util.SortedMap<GenPolynomial<C>,java.lang.Long>	squarefreeFactors(GenPolynomial<C> P) GenPolynomial squarefree factorization.
C	squarefreePart(C P) Coefficients greatest squarefree divisor.
abstract GenPolynomial<C>	squarefreePart(GenPolynomial<C> P) GenPolynomial greatest squarefree divisor.

Methods inherited from class java.lang.Object

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

Field Detail

engine

protected final GreatestCommonDivisorAbstract<C extends GcdRingElem<C>> engine

GCD engine for respective base coefficients.

Constructor Detail

SquarefreeAbstract

public SquarefreeAbstract(GreatestCommonDivisorAbstract<C> engine)

Constructor.

Method Detail

baseSquarefreePart

public abstract GenPolynomial<C> baseSquarefreePart(GenPolynomial<C> P)

GenPolynomial polynomial greatest squarefree divisor.

Parameters:

P - GenPolynomial.

Returns:

squarefree(pp(P)).

baseSquarefreeFactors

public abstract java.util.SortedMap<GenPolynomial<C>,java.lang.Long> baseSquarefreeFactors(GenPolynomial<C> A)

GenPolynomial polynomial squarefree factorization.

Parameters:

A - GenPolynomial.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i^{e_i} and p_i squarefree.

recursiveUnivariateSquarefreePart

public abstract GenPolynomial<GenPolynomial<C>> recursiveUnivariateSquarefreePart(GenPolynomial<GenPolynomial<C>> P)

GenPolynomial recursive polynomial greatest squarefree divisor.

Parameters:

P - recursive univariate GenPolynomial.

Returns:

squarefree(pp(P)).

recursiveUnivariateSquarefreeFactors

public abstract java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long> recursiveUnivariateSquarefreeFactors(GenPolynomial<GenPolynomial<C>> P)

GenPolynomial recursive univariate polynomial squarefree factorization.

Parameters:

P - recursive univariate GenPolynomial.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i^{e_i} and p_i squarefree.

squarefreePart

public abstract GenPolynomial<C> squarefreePart(GenPolynomial<C> P)

GenPolynomial greatest squarefree divisor.

Specified by:

squarefreePart in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

squarefree(P) a primitive respectively monic polynomial.

isSquarefree

public boolean isSquarefree(GenPolynomial<C> P)

GenPolynomial test if is squarefree.

Specified by:

isSquarefree in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

true if P is squarefree, else false.

isSquarefree

public boolean isSquarefree(java.util.List<GenPolynomial<C>> L)

GenPolynomial list test if squarefree.

Specified by:

isSquarefree in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

L - list of GenPolynomial.

Returns:

true if each P in L is squarefree, else false.

isRecursiveSquarefree

public boolean isRecursiveSquarefree(GenPolynomial<GenPolynomial<C>> P)

Recursive GenPolynomial test if is squarefree.

Parameters:

P - recursive univariate GenPolynomial.

Returns:

true if P is squarefree, else false.

squarefreeFactors

public abstract java.util.SortedMap<GenPolynomial<C>,java.lang.Long> squarefreeFactors(GenPolynomial<C> P)

GenPolynomial squarefree factorization.

Specified by:

squarefreeFactors in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i^{e_i} and p_i squarefree.

coPrimeSquarefree

public java.util.List<GenPolynomial<C>> coPrimeSquarefree(java.util.List<GenPolynomial<C>> A)

GenPolynomial squarefree and co-prime list.

Specified by:

coPrimeSquarefree in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

A - list of GenPolynomials.

Returns:

B with gcd(b,c) = 1 for all b != c in B and for all non-constant a in A there exists b in B with b|a and each b in B is squarefree. B does not contain zero or constant polynomials.

coPrimeSquarefree

public java.util.List<GenPolynomial<C>> coPrimeSquarefree(GenPolynomial<C> a,
                                                          java.util.List<GenPolynomial<C>> P)

GenPolynomial squarefree and co-prime list.

Specified by:

coPrimeSquarefree in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

a - polynomial.

P - squarefree co-prime list of GenPolynomials.

Returns:

B with gcd(b,c) = 1 for all b != c in B and for non-constant a there exists b in P with b|a. B does not contain zero or constant polynomials.

isCoPrimeSquarefree

public boolean isCoPrimeSquarefree(java.util.List<GenPolynomial<C>> B)

Test if list of GenPolynomials is squarefree and co-prime.

Specified by:

isCoPrimeSquarefree in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

B - list of GenPolynomials.

Returns:

true, if for all b != c in B gcd(b,c) = 1 and each b in B is squarefree, else false.

normalizeFactorization

public java.util.SortedMap<GenPolynomial<C>,java.lang.Long> normalizeFactorization(java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F)

Normalize factorization. p'_i > 0 for i > 1 and p'_1 != 1 if k > 1.

Parameters:

F - = [p_1->e_1;, ..., p_k->e_k].

Returns:

F' = [p'_1->e_1, ..., p'_k->e_k].

isFactorization

public boolean isFactorization(GenPolynomial<C> P,
                               java.util.List<GenPolynomial<C>> F)

GenPolynomial is (squarefree) factorization.

Specified by:

isFactorization in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

F - = [p_1,...,p_k].

Returns:

true if P = prod_{i=1,...,r} p_i, else false.

factorCount

public long factorCount(java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F)

Count number of factors in a (squarefree) factorization.

Parameters:

F - = [p_1 -> e_1, ..., p_k -> e_k].

Returns:

sum_{i=1,...,k} e_i.

isFactorization

public boolean isFactorization(GenPolynomial<C> P,
                               java.util.SortedMap<GenPolynomial<C>,java.lang.Long> F)

GenPolynomial is (squarefree) factorization.

Specified by:

isFactorization in interface Squarefree<C extends GcdRingElem<C>>

Parameters:

P - GenPolynomial.

F - = [p_1 -> e_1, ..., p_k -> e_k].

Returns:

true if P = prod_{i=1,...,k} p_i**e_i, else false.

isRecursiveFactorization

public boolean isRecursiveFactorization(GenPolynomial<GenPolynomial<C>> P,
                                        java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long> F)

GenPolynomial is (squarefree) factorization.

Parameters:

P - GenPolynomial.

F - = [p_1 -> e_1, ..., p_k -> e_k].

Returns:

true if P = prod_{i=1,...,k} p_i**e_i, else false.

recursiveSquarefreePart

public GenPolynomial<GenPolynomial<C>> recursiveSquarefreePart(GenPolynomial<GenPolynomial<C>> P)

GenPolynomial recursive polynomial greatest squarefree divisor.

Parameters:

P - recursive GenPolynomial.

Returns:

squarefree(pp(P)).

recursiveSquarefreeFactors

public java.util.SortedMap<GenPolynomial<GenPolynomial<C>>,java.lang.Long> recursiveSquarefreeFactors(GenPolynomial<GenPolynomial<C>> P)

GenPolynomial recursive polynomial squarefree factorization.

Parameters:

P - recursive GenPolynomial.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i^{e_i} and p_i squarefree.

basePartialFraction

public java.util.List<java.util.List<GenPolynomial<C>>> basePartialFraction(GenPolynomial<C> A,
                                                                            java.util.SortedMap<GenPolynomial<C>,java.lang.Long> D)

Univariate GenPolynomial partial fraction decomposition.

Parameters:

A - univariate GenPolynomial.

D - sorted map [d_1 -> e_1, ..., d_k -> e_k] with d_i squarefree.

Returns:

[ [Ai0, Ai1,..., Aie_i], i=0,...,k ] with A/prod(D) = A0 + sum( sum ( Aij/di^j ) ) with deg(Aij) < deg(di).

isBasePartialFraction

public boolean isBasePartialFraction(GenPolynomial<C> A,
                                     java.util.SortedMap<GenPolynomial<C>,java.lang.Long> D,
                                     java.util.List<java.util.List<GenPolynomial<C>>> F)

Test for Univariate GenPolynomial partial fraction decomposition.

Parameters:

A - univariate GenPolynomial.

D - sorted map [d_1 -> e_1, ..., d_k -> e_k] with d_i squarefree.

F - a list of lists [ [Ai0, Ai1,..., Aie_i], i=0,...,k ]

Returns:

true, if A/prod(D) = A0 + sum( sum ( Aij/di^j ) ), else false.

squarefreePart

public C squarefreePart(C P)

Coefficients greatest squarefree divisor.

Parameters:

P - coefficient.

Returns:

squarefree part of P.

squarefreeFactors

public abstract java.util.SortedMap<C,java.lang.Long> squarefreeFactors(C P)

Coefficients squarefree factorization.

Parameters:

P - coefficient.

Returns:

[p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i^{e_i} and p_i squarefree.