Package edu.jas.ufd

Class PolyUfdUtil

java.lang.Object

edu.jas.ufd.PolyUfdUtil

public class PolyUfdUtil
extends java.lang.Object

Polynomial ufd utilities. For example conversion between different representations and Kronecker substitution.

Author:

Heinz Kredel

Constructor Summary

Constructors

Constructor	Description
PolyUfdUtil()

Method Summary

Modifier and Type	Method	Description
static <C extends GcdRingElem<C>> AlgebraicNumberRing<C>	algebraicNumberField(RingFactory<C> cfac, int degree)	Construct an algebraic number field of degree d.
static <C extends GcdRingElem<C>> GenPolynomial<C>	backSubstituteKronecker(GenPolynomialRing<C> fac, GenPolynomial<C> A, long d)	Kronecker back substitution.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>>	backSubstituteKronecker(GenPolynomialRing<C> fac, java.util.List<GenPolynomial<C>> A, long d)	Kronecker back substitution.
static <C extends GcdRingElem<C>> void	ensureFieldProperty(AlgebraicNumberRing<C> afac)	Ensure that the field property is determined.
static <C extends GcdRingElem<C>> java.util.SortedMap<Quotient<C>,java.lang.Long>	factors(Quotient<C> A)	Factors of Quotient rational function.
static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>	fromIntegerCoefficients(GenPolynomialRing<GenPolynomial<C>> fac, GenPolynomial<GenPolynomial<BigInteger>> A)	From BigInteger coefficients.
static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>>	fromIntegerCoefficients(GenPolynomialRing<GenPolynomial<C>> fac, java.util.List<GenPolynomial<GenPolynomial<BigInteger>>> L)	From BigInteger coefficients.
static GenPolynomial<GenPolynomial<BigInteger>>	integerFromRationalCoefficients(GenPolynomialRing<GenPolynomial<BigInteger>> fac, GenPolynomial<GenPolynomial<BigRational>> A)	BigInteger from BigRational coefficients.
static java.util.List<GenPolynomial<GenPolynomial<BigInteger>>>	integerFromRationalCoefficients(GenPolynomialRing<GenPolynomial<BigInteger>> fac, java.util.List<GenPolynomial<GenPolynomial<BigRational>>> L)	BigInteger from BigRational coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>>	integralFromQuotientCoefficients(GenPolynomialRing<GenPolynomial<C>> fac, GenPolynomial<Quotient<C>> A)	Integral polynomial from rational function coefficients.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>>	integralFromQuotientCoefficients(GenPolynomialRing<GenPolynomial<C>> fac, java.util.Collection<GenPolynomial<Quotient<C>>> L)	Integral polynomial from rational function coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>>	introduceLowerVariable(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<C> A)	Introduce lower variable.
static <C extends GcdRingElem<C>> boolean	isFactorization(Quotient<C> P, java.util.SortedMap<Quotient<C>,java.lang.Long> F)	Quotient is (squarefree) factorization.
static <C extends GcdRingElem<C>> GenPolynomial<C>	norm(GenPolynomial<AlgebraicNumber<C>> A)	Norm of a polynomial with AlgebraicNumber coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<C>	norm(GenPolynomial<AlgebraicNumber<C>> A, long k)	Norm of a polynomial with AlgebraicNumber coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<Quotient<C>>	quotientFromIntegralCoefficients(GenPolynomialRing<Quotient<C>> fac, GenPolynomial<GenPolynomial<C>> A)	Rational function from integral polynomial coefficients.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Quotient<C>>>	quotientFromIntegralCoefficients(GenPolynomialRing<Quotient<C>> fac, java.util.Collection<GenPolynomial<GenPolynomial<C>>> L)	Rational function from integral polynomial coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<C>	randomIrreduciblePolynomial(RingFactory<C> cfac, int degree)	Construct a random irreducible univariate polynomial of degree d.
static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>	substituteConvertToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<C> A, long k)	Convert to AlgebraicNumber coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>>	substituteFromAlgebraicCoefficients(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<AlgebraicNumber<C>> A, long k)	From AlgebraicNumber coefficients.
static <C extends GcdRingElem<C>> GenPolynomial<C>	substituteKronecker(GenPolynomial<C> A)	Kronecker substitution.
static <C extends GcdRingElem<C>> GenPolynomial<C>	substituteKronecker(GenPolynomial<C> A, long d)	Kronecker substitution.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>>	substituteKronecker(java.util.List<GenPolynomial<C>> A, int d)	Kronecker substitution.

Methods inherited from class java.lang.Object

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

Constructor Detail

PolyUfdUtil

public PolyUfdUtil()

Method Detail

factors

public static <C extends GcdRingElem<C>> java.util.SortedMap<Quotient<C>,java.lang.Long> factors(Quotient<C> A)

Factors of Quotient rational function.

Parameters:

A - rational function to be factored.

Returns:

list of irreducible rational function parts.

isFactorization

public static <C extends GcdRingElem<C>> boolean isFactorization(Quotient<C> P,
                                                                 java.util.SortedMap<Quotient<C>,java.lang.Long> F)

Quotient is (squarefree) factorization.

Parameters:

P - Quotient.

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

Returns:

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

integralFromQuotientCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>> integralFromQuotientCoefficients(GenPolynomialRing<GenPolynomial<C>> fac,
                                                                                                          GenPolynomial<Quotient<C>> A)

Integral polynomial from rational function coefficients. Represent as polynomial with integral polynomial coefficients by multiplication with the lcm of the numerators of the rational function coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with rational function coefficients to be converted.

Returns:

polynomial with integral polynomial coefficients.

integralFromQuotientCoefficients

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>> integralFromQuotientCoefficients(GenPolynomialRing<GenPolynomial<C>> fac,
                                                                                                                          java.util.Collection<GenPolynomial<Quotient<C>>> L)

Integral polynomial from rational function coefficients. Represent as polynomial with integral polynomial coefficients by multiplication with the lcm of the numerators of the rational function coefficients.

Parameters:

fac - result polynomial factory.

L - list of polynomial with rational function coefficients to be converted.

Returns:

list of polynomials with integral polynomial coefficients.

quotientFromIntegralCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<Quotient<C>> quotientFromIntegralCoefficients(GenPolynomialRing<Quotient<C>> fac,
                                                                                                     GenPolynomial<GenPolynomial<C>> A)

Rational function from integral polynomial coefficients. Represent as polynomial with type Quotient coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with integral polynomial coefficients to be converted.

Returns:

polynomial with type Quotient coefficients.

quotientFromIntegralCoefficients

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Quotient<C>>> quotientFromIntegralCoefficients(GenPolynomialRing<Quotient<C>> fac,
                                                                                                                     java.util.Collection<GenPolynomial<GenPolynomial<C>>> L)

Rational function from integral polynomial coefficients. Represent as polynomial with type Quotient coefficients.

Parameters:

fac - result polynomial factory.

L - list of polynomials with integral polynomial coefficients to be converted.

Returns:

list of polynomials with type Quotient coefficients.

fromIntegerCoefficients

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> fromIntegerCoefficients(GenPolynomialRing<GenPolynomial<C>> fac,
                                                                                              GenPolynomial<GenPolynomial<BigInteger>> A)

From BigInteger coefficients. Represent as polynomial with type GenPolynomial<C> coefficients, e.g. ModInteger or BigRational.

Parameters:

fac - result polynomial factory.

A - polynomial with GenPolynomial<BigInteger> coefficients to be converted.

Returns:

polynomial with type GenPolynomial<C> coefficients.

fromIntegerCoefficients

public static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>> fromIntegerCoefficients(GenPolynomialRing<GenPolynomial<C>> fac,
                                                                                                              java.util.List<GenPolynomial<GenPolynomial<BigInteger>>> L)

From BigInteger coefficients. Represent as polynomial with type GenPolynomial<C> coefficients, e.g. ModInteger or BigRational.

Parameters:

fac - result polynomial factory.

L - polynomial list with GenPolynomial<BigInteger> coefficients to be converted.

Returns:

polynomial list with polynomials with type GenPolynomial<C> coefficients.

integerFromRationalCoefficients

public static GenPolynomial<GenPolynomial<BigInteger>> integerFromRationalCoefficients(GenPolynomialRing<GenPolynomial<BigInteger>> fac,
                                                                                       GenPolynomial<GenPolynomial<BigRational>> A)

BigInteger from BigRational coefficients. Represent as polynomial with type GenPolynomial<BigInteger> coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with GenPolynomial<BigRational> coefficients to be converted.

Returns:

polynomial with type GenPolynomial<BigInteger> coefficients.

integerFromRationalCoefficients

public static java.util.List<GenPolynomial<GenPolynomial<BigInteger>>> integerFromRationalCoefficients(GenPolynomialRing<GenPolynomial<BigInteger>> fac,
                                                                                                       java.util.List<GenPolynomial<GenPolynomial<BigRational>>> L)

BigInteger from BigRational coefficients. Represent as polynomial with type GenPolynomial<BigInteger> coefficients.

Parameters:

fac - result polynomial factory.

L - polynomial list with GenPolynomial<BigRational> coefficients to be converted.

Returns:

polynomial list with polynomials with type GenPolynomial<BigInteger> coefficients.

introduceLowerVariable

public static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>> introduceLowerVariable(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                                GenPolynomial<C> A)

Introduce lower variable. Represent as polynomial with type GenPolynomial<C> coefficients.

Parameters:

rfac - result polynomial factory.

A - polynomial to be extended.

Returns:

polynomial with type GenPolynomial<C> coefficients.

substituteFromAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>> substituteFromAlgebraicCoefficients(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                                             GenPolynomial<AlgebraicNumber<C>> A,
                                                                                                             long k)

From AlgebraicNumber coefficients. Represent as polynomial with type GenPolynomial<C> coefficients, e.g. ModInteger or BigRational.

Parameters:

rfac - result polynomial factory.

A - polynomial with AlgebraicNumber coefficients to be converted.

k - for (y-k x) substitution.

Returns:

polynomial with type GenPolynomial<C> coefficients.

substituteConvertToAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> substituteConvertToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac,
                                                                                                                    GenPolynomial<C> A,
                                                                                                                    long k)

Convert to AlgebraicNumber coefficients. Represent as polynomial with AlgebraicNumber coefficients, C is e.g. ModInteger or BigRational.

Parameters:

pfac - result polynomial factory.

A - polynomial with GenPolynomial<BigInteger> coefficients to be converted.

k - for (y-k x) substitution.

Returns:

polynomial with AlgebraicNumber<C> coefficients.

norm

public static <C extends GcdRingElem<C>> GenPolynomial<C> norm(GenPolynomial<AlgebraicNumber<C>> A,
                                                               long k)

Norm of a polynomial with AlgebraicNumber coefficients.

Parameters:

A - uni or multivariate polynomial from GenPolynomial<AlgebraicNumber<C>>.

k - for (y - k x) substitution.

Returns:

norm(A) = res_x(A(x,y),m(x)) in GenPolynomialRing<C>.

norm

public static <C extends GcdRingElem<C>> GenPolynomial<C> norm(GenPolynomial<AlgebraicNumber<C>> A)

Norm of a polynomial with AlgebraicNumber coefficients.

Parameters:

A - polynomial from GenPolynomial<AlgebraicNumber<C>>.

Returns:

norm(A) = resultant_x( A(x,y), m(x) ) in K[y].

ensureFieldProperty

public static <C extends GcdRingElem<C>> void ensureFieldProperty(AlgebraicNumberRing<C> afac)

Ensure that the field property is determined. Checks if modul is irreducible and modifies the algebraic number ring.

Parameters:

afac - algebraic number ring.

randomIrreduciblePolynomial

public static <C extends GcdRingElem<C>> GenPolynomial<C> randomIrreduciblePolynomial(RingFactory<C> cfac,
                                                                                      int degree)

Construct a random irreducible univariate polynomial of degree d.

Parameters:

cfac - coefficient ring.

degree - of random polynomial.

Returns:

irreducible univariate polynomial.

algebraicNumberField

public static <C extends GcdRingElem<C>> AlgebraicNumberRing<C> algebraicNumberField(RingFactory<C> cfac,
                                                                                     int degree)

Construct an algebraic number field of degree d. Uses a random irreducible polynomial of degree d as modulus of the algebraic number ring.

Parameters:

cfac - coefficient ring.

degree - of random polynomial.

Returns:

algebraic number field.

substituteKronecker

public static <C extends GcdRingElem<C>> GenPolynomial<C> substituteKronecker(GenPolynomial<C> A)

Kronecker substitution. Substitute x_i by x**d**(i-1) to construct a univariate polynomial.

Parameters:

A - polynomial to be converted.

Returns:

a univariate polynomial.

substituteKronecker

public static <C extends GcdRingElem<C>> GenPolynomial<C> substituteKronecker(GenPolynomial<C> A,
                                                                              long d)

Kronecker substitution. Substitute x_i by x**d**(i-1) to construct a univariate polynomial.

Parameters:

A - polynomial to be converted.

Returns:

a univariate polynomial.

substituteKronecker

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>> substituteKronecker(java.util.List<GenPolynomial<C>> A,
                                                                                              int d)

Kronecker substitution. Substitute x_i by x**d**(i-1) to construct univariate polynomials.

Parameters:

A - list of polynomials to be converted.

Returns:

a list of univariate polynomials.

backSubstituteKronecker

public static <C extends GcdRingElem<C>> GenPolynomial<C> backSubstituteKronecker(GenPolynomialRing<C> fac,
                                                                                  GenPolynomial<C> A,
                                                                                  long d)

Kronecker back substitution. Substitute x**d**(i-1) to x_i to construct a multivariate polynomial.

Parameters:

A - polynomial to be converted.

fac - result polynomial factory.

Returns:

a multivariate polynomial.

backSubstituteKronecker

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<C>> backSubstituteKronecker(GenPolynomialRing<C> fac,
                                                                                                  java.util.List<GenPolynomial<C>> A,
                                                                                                  long d)

Kronecker back substitution. Substitute x**d**(i-1) to x_i to construct multivariate polynomials.

Parameters:

A - list of polynomials to be converted.

fac - result polynomial factory.

Returns:

a list of multivariate polynomials.