edu.jas.ufd

Class PolyUfdUtil

java.lang.Object

edu.jas.ufd.PolyUfdUtil

public class PolyUfdUtil
extends java.lang.Object

Polynomial ufd utilities, like conversion between different representations and Hensel lifting.

Constructor Summary

Constructors

Constructor and Description
PolyUfdUtil()

Method Summary

Modifier and Type	Method and Description
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 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 <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>>	integralFromQuotientCoefficients(GenPolynomialRing<GenPolynomial<C>> fac, GenPolynomial<Quotient<C>> A) 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>> 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>> 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<Quotient<C>>	quotientFromIntegralCoefficients(GenPolynomialRing<Quotient<C>> fac, GenPolynomial<GenPolynomial<C>> A) Rational function from integral polynomial coefficients.
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

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

Constructor Detail

PolyUfdUtil

public PolyUfdUtil()

Method Detail

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.

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 - 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.

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 a 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 a multivariate polynomials.

Parameters:

A - list of polynomials to be converted.

fac - result polynomial factory.

Returns:

a list of multivariate polynomials.