edu.jas.gbufd

Class PolyGBUtil

java.lang.Object

edu.jas.gbufd.PolyGBUtil

public class PolyGBUtil
extends java.lang.Object

Package gbufd utilities.

Constructor Summary

Constructors

Constructor and Description
PolyGBUtil()

Method Summary

Modifier and Type	Method and Description
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>> 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>> 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>> GenPolynomial<C>	syzGcd(GenPolynomialRing<C> r, GenPolynomial<C> n, GenPolynomial<C> d) Greatest common divisor.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	syzGcd(GenSolvablePolynomialRing<C> r, GenSolvablePolynomial<C> n, GenSolvablePolynomial<C> d) Greatest common divisor via least common multiple.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[]	syzGcdCofactors(GenSolvablePolynomialRing<C> r, GenSolvablePolynomial<C> n, GenSolvablePolynomial<C> d) Greatest common divisor and cofactors via least common multiple and reduction.
static <C extends GcdRingElem<C>> GenPolynomial<C>	syzLcm(GenPolynomialRing<C> r, GenPolynomial<C> n, GenPolynomial<C> d) Least common multiple.
static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>	syzLcm(GenSolvablePolynomialRing<C> r, GenSolvablePolynomial<C> n, GenSolvablePolynomial<C> d) Least common multiple via ideal intersection.
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

equals, 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)

syzLcm

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> syzLcm(GenSolvablePolynomialRing<C> r,
                                                                         GenSolvablePolynomial<C> n,
                                                                         GenSolvablePolynomial<C> d)

Least common multiple via ideal intersection.

Parameters:

r - solvable polynomial ring.

n - first solvable polynomial.

d - second solvable polynomial.

Returns:

lcm(n,d)

syzGcd

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C> syzGcd(GenSolvablePolynomialRing<C> r,
                                                                         GenSolvablePolynomial<C> n,
                                                                         GenSolvablePolynomial<C> d)

Greatest common divisor via least common multiple.

Parameters:

r - solvable polynomial ring.

n - first solvable polynomial.

d - second solvable polynomial.

Returns:

gcd(n,d)

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 ]

syzGcdCofactors

public static <C extends GcdRingElem<C>> GenSolvablePolynomial<C>[] syzGcdCofactors(GenSolvablePolynomialRing<C> r,
                                                                                    GenSolvablePolynomial<C> n,
                                                                                    GenSolvablePolynomial<C> d)

Greatest common divisor and cofactors via least common multiple and reduction.

Parameters:

r - solvable polynomial ring.

n - first solvable polynomial.

d - second solvable polynomial.

Returns:

[ g=gcd(n,d), n/g, d/g ]

syzLcm

public static <C extends GcdRingElem<C>> GenPolynomial<C> syzLcm(GenPolynomialRing<C> r,
                                                                 GenPolynomial<C> n,
                                                                 GenPolynomial<C> d)

Least common multiple. Just for fun, is not efficient.

Parameters:

r - polynomial ring.

n - first polynomial.

d - second polynomial.

Returns:

lcm(n,d)

syzGcd

public static <C extends GcdRingElem<C>> GenPolynomial<C> syzGcd(GenPolynomialRing<C> r,
                                                                 GenPolynomial<C> n,
                                                                 GenPolynomial<C> d)

Greatest common divisor. Just for fun, is not efficient.

Parameters:

r - polynomial ring.

n - first polynomial.

d - second polynomial.

Returns:

gcd(n,d)