Package edu.jas.gbufd

Class PolyGBUtil

java.lang.Object

edu.jas.gbufd.PolyGBUtil

public class PolyGBUtil
extends java.lang.Object

Package gbufd utilities.

Author:

Heinz Kredel

Constructor Summary

Constructors

Constructor	Description
PolyGBUtil()

Method Summary

Modifier and Type	Method	Description
static <C extends GcdRingElem<C>> GenPolynomial<C>	chineseRemainderTheorem(java.util.List<java.util.List<GenPolynomial<C>>> F, java.util.List<GenPolynomial<C>> A)	Chinese remainder theorem.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<GenPolynomial<C>>>	coefficientPseudoRemainder(GenPolynomial<GenPolynomial<GenPolynomial<C>>> P, GenPolynomial<GenPolynomial<C>> A)	Polynomial leading coefficient pseudo remainder.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	coefficientPseudoRemainderBase(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<C> A)	Polynomial leading coefficient pseudo remainder, base case.
static <C extends GcdRingElem<C>> GenPolynomial<C>	CRTInterpolation(GenPolynomialRing<C> fac, java.util.List<java.util.List<C>> E, java.util.List<C> V)	Chinese remainder theorem, interpolation.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>>	intersect(GenPolynomialRing<C> pfac, java.util.List<GenPolynomial<C>> A, java.util.List<GenPolynomial<C>> B)	Intersection.
static <C extends GcdRingElem<C>> java.util.List<GenSolvablePolynomial<C>>	intersect(GenSolvablePolynomialRing<C> pfac, java.util.List<GenSolvablePolynomial<C>> A, java.util.List<GenSolvablePolynomial<C>> B)	Intersection.
static <C extends GcdRingElem<C>> java.util.List<GenWordPolynomial<C>>	intersect(GenWordPolynomialRing<C> pfac, java.util.List<GenWordPolynomial<C>> A, java.util.List<GenWordPolynomial<C>> B)	Intersection.
static <C extends GcdRingElem<C>> java.util.List<GenWordPolynomial<C>>	intersect(GenWordPolynomialRing<C> pfac, java.util.List<GenWordPolynomial<C>> A, java.util.List<GenWordPolynomial<C>> B, WordGroebnerBaseAbstract<C> bb)	Intersection.
static <C extends GcdRingElem<C>> boolean	isChineseRemainder(java.util.List<java.util.List<GenPolynomial<C>>> F, java.util.List<GenPolynomial<C>> A, GenPolynomial<C> h)	Is Chinese remainder.
static <C extends GcdRingElem<C>> boolean	isResultant(GenPolynomial<C> A, GenPolynomial<C> B, GenPolynomial<C> r)	Test for resultant.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[]	quotientRemainder(GenSolvablePolynomial<C> n, GenSolvablePolynomial<C> d)	Solvable quotient and remainder via reduction.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>>	subRing(java.util.List<GenPolynomial<C>> A)	Subring generators.
static <C extends GcdRingElem<C>> boolean	subRingAndMember(java.util.List<GenPolynomial<C>> A, GenPolynomial<C> g)	Subring and membership test.
static <C extends GcdRingElem<C>> boolean	subRingMember(java.util.List<GenPolynomial<C>> A, GenPolynomial<C> g)	Subring membership.
static <C extends RingElem<C>> GenPolynomial<C>	topCoefficientPseudoRemainder(java.util.List<GenPolynomial<C>> A, GenPolynomial<C> P)	Top coefficient pseudo remainder of the leading coefficient of P wrt A in the main variables.
static <C extends RingElem<C>> GenPolynomial<C>	topPseudoRemainder(java.util.List<GenPolynomial<C>> A, GenPolynomial<C> P)	Top pseudo reduction wrt the main variables.
static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>	zeroDegrees(java.util.List<GenPolynomial<C>> A)	Extract polynomials with degree zero in the main variable.

Methods inherited from class java.lang.Object

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

Constructor Detail

PolyGBUtil

public PolyGBUtil()

Method Detail

isResultant

public static <C extends GcdRingElem<C>> boolean isResultant(GenPolynomial<C> A,
                                                             GenPolynomial<C> B,
                                                             GenPolynomial<C> r)

Test for resultant.

Parameters:

A - generic polynomial.

B - generic polynomial.

r - generic polynomial.

Returns:

true if res(A,B) isContained in ideal(A,B), else false.

topPseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<C> topPseudoRemainder(java.util.List<GenPolynomial<C>> A,
                                                                          GenPolynomial<C> P)

Top pseudo reduction wrt the main variables.

Parameters:

P - generic polynomial.

A - list of generic polynomials sorted according to appearing main variables.

Returns:

top pseudo remainder of P wrt. A for the appearing variables.

topCoefficientPseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<C> topCoefficientPseudoRemainder(java.util.List<GenPolynomial<C>> A,
                                                                                     GenPolynomial<C> P)

Top coefficient pseudo remainder of the leading coefficient of P wrt A in the main variables.

Parameters:

P - generic polynomial in n+1 variables.

A - list of generic polynomials in n variables sorted according to appearing main variables.

Returns:

pseudo remainder of the leading coefficient of P wrt A.

coefficientPseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<GenPolynomial<C>>> coefficientPseudoRemainder(GenPolynomial<GenPolynomial<GenPolynomial<C>>> P,
                                                                                                                GenPolynomial<GenPolynomial<C>> A)

Polynomial leading coefficient pseudo remainder.

Parameters:

P - generic polynomial in n+1 variables.

A - generic polynomial in n variables.

Returns:

pseudo remainder of the leading coefficient of P wrt A, with ldcf(A)^m' P = quotient * A + remainder.

coefficientPseudoRemainderBase

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> coefficientPseudoRemainderBase(GenPolynomial<GenPolynomial<C>> P,
                                                                                                     GenPolynomial<C> A)

Polynomial leading coefficient pseudo remainder, base case.

Parameters:

P - generic polynomial in 1+1 variables.

A - generic polynomial in 1 variable.

Returns:

pseudo remainder of the leading coefficient of P wrt. A, with ldcf(A)^m' P = quotient * A + remainder.

zeroDegrees

public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> zeroDegrees(java.util.List<GenPolynomial<C>> A)

Extract polynomials with degree zero in the main variable.

Parameters:

A - list of generic polynomials in n variables.

Returns:

Z = [a_i] with deg(a_i,x_n) = 0 and in n-1 variables.

intersect

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>> intersect(GenPolynomialRing<C> pfac,
                                                                                    java.util.List<GenPolynomial<C>> A,
                                                                                    java.util.List<GenPolynomial<C>> B)

Intersection. Generators for the intersection of ideals.

Parameters:

pfac - polynomial ring

A - list of polynomials

B - list of polynomials

Returns:

generators for (A \cap B)

intersect

public static <C extends GcdRingElem<C>> java.util.List<GenSolvablePolynomial<C>> intersect(GenSolvablePolynomialRing<C> pfac,
                                                                                            java.util.List<GenSolvablePolynomial<C>> A,
                                                                                            java.util.List<GenSolvablePolynomial<C>> B)

Intersection. Generators for the intersection of ideals.

Parameters:

pfac - solvable polynomial ring

A - list of polynomials

B - list of polynomials

Returns:

generators for (A \cap B)

intersect

public static <C extends GcdRingElem<C>> java.util.List<GenWordPolynomial<C>> intersect(GenWordPolynomialRing<C> pfac,
                                                                                        java.util.List<GenWordPolynomial<C>> A,
                                                                                        java.util.List<GenWordPolynomial<C>> B)

Intersection. Generators for the intersection of word ideals.

Parameters:

pfac - word polynomial ring

A - list of word polynomials

B - list of word polynomials

Returns:

generators for (A \cap B) if it exists

intersect

public static <C extends GcdRingElem<C>> java.util.List<GenWordPolynomial<C>> intersect(GenWordPolynomialRing<C> pfac,
                                                                                        java.util.List<GenWordPolynomial<C>> A,
                                                                                        java.util.List<GenWordPolynomial<C>> B,
                                                                                        WordGroebnerBaseAbstract<C> bb)

Intersection. Generators for the intersection of word ideals.

Parameters:

pfac - word polynomial ring

A - list of word polynomials

B - list of word polynomials

bb - Groebner Base engine

Returns:

generators for (A \cap B) if it exists

quotientRemainder

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[] quotientRemainder(GenSolvablePolynomial<C> n,
                                                                                      GenSolvablePolynomial<C> d)

Solvable quotient and remainder via reduction.

Parameters:

n - first solvable polynomial.

d - second solvable polynomial.

Returns:

[ n/d, n - (n/d)*d ]

subRing

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>> subRing(java.util.List<GenPolynomial<C>> A)

Subring generators.

Parameters:

A - list of polynomials in n variables.

Returns:

a Groebner base of polynomials in m > n variables generating the subring of K[A].

subRingMember

public static <C extends GcdRingElem<C>> boolean subRingMember(java.util.List<GenPolynomial<C>> A,
                                                               GenPolynomial<C> g)

Subring membership.

Parameters:

A - Groebner base of polynomials in m > n variables generating the subring of elements of K[A].

g - polynomial in n variables.

Returns:

true, if g \in K[A], else false.

subRingAndMember

public static <C extends GcdRingElem<C>> boolean subRingAndMember(java.util.List<GenPolynomial<C>> A,
                                                                  GenPolynomial<C> g)

Subring and membership test.

Parameters:

A - list of polynomials in n variables.

g - polynomial in n variables.

Returns:

true, if g \in K[A], else false.

chineseRemainderTheorem

public static <C extends GcdRingElem<C>> GenPolynomial<C> chineseRemainderTheorem(java.util.List<java.util.List<GenPolynomial<C>>> F,
                                                                                  java.util.List<GenPolynomial<C>> A)

Chinese remainder theorem.

Parameters:

F - = ( F_i ) list of list of polynomials in n variables.

A - = ( f_i ) list of polynomials in n variables.

Returns:

p \in \Cap_i (f_i + ideal(F_i)) if it exists, else null.

isChineseRemainder

public static <C extends GcdRingElem<C>> boolean isChineseRemainder(java.util.List<java.util.List<GenPolynomial<C>>> F,
                                                                    java.util.List<GenPolynomial<C>> A,
                                                                    GenPolynomial<C> h)

Is Chinese remainder.

Parameters:

F - = ( F_i ) list of list of polynomials in n variables.

A - = ( f_i ) list of polynomials in n variables.

h - polynomial in n variables.

Returns:

true if h \in \Cap_i (f_i + ideal(F_i)), else false.

CRTInterpolation

public static <C extends GcdRingElem<C>> GenPolynomial<C> CRTInterpolation(GenPolynomialRing<C> fac,
                                                                           java.util.List<java.util.List<C>> E,
                                                                           java.util.List<C> V)

Chinese remainder theorem, interpolation.

Parameters:

fac - polynomial ring over K in n variables.

E - = ( E_i ), E_i = ( e_ij ) list of list of elements of K, the evaluation points.

V - = ( f_i ) list of elements of K, the evaluation values.

Returns:

p \in K[X1,...,Xn], with p(E_i) = f_i, if it exists, else null.