Package cc.redberry.rings

Class Rings

java.lang.Object

cc.redberry.rings.Rings

public final class Rings
extends Object

Common rings.

Since:

1.0

Field Summary

Fields

Modifier and Type	Field	Description
static AlgebraicNumberField<UnivariatePolynomial<BigInteger>>	GaussianIntegers	Ring of Gaussian integers (integer complex numbers).
static AlgebraicNumberField<UnivariatePolynomial<Rational<BigInteger>>>	GaussianRationals	Field of Gaussian rationals (rational complex numbers).
static Rationals<BigInteger>	Q	Field of rationals (Q)
static UnivariateRing<UnivariatePolynomial<Rational<BigInteger>>>	UnivariateRingQ	Ring of univariate polynomials over rationals (Q[x])
static UnivariateRing<UnivariatePolynomial<BigInteger>>	UnivariateRingZ	Ring of univariate polynomials over integers (Z[x])
static Integers	Z	Ring of integers (Z)

Method Summary

Modifier and Type	Method	Description
static <Poly extends IUnivariatePolynomial<Poly>> AlgebraicNumberField<Poly>	AlgebraicNumberField(Poly minimalPoly)	Algebraic number field generated by the specified minimal polynomial
static <E> Rationals<E>	Frac(Ring<E> ring)	Ring of rational functions over specified ring
static <E> AlgebraicNumberField<UnivariatePolynomial<E>>	GaussianNumbers(Ring<E> ring)	Gaussian numbers for a given ring (that is ring adjoined with imaginary unit)
static FiniteField<UnivariatePolynomialZp64>	GF(long prime, int exponent)	Galois field with the cardinality prime ^ exponent (with prime < 2^63).
static FiniteField<UnivariatePolynomial<BigInteger>>	GF(BigInteger prime, int exponent)	Galois field with the cardinality prime ^ exponent for arbitrary large prime
static <Poly extends IUnivariatePolynomial<Poly>> FiniteField<Poly>	GF(Poly irreducible)	Galois field with the specified minimal polynomial.
static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly>	MultipleFieldExtension(sPoly... minimalPolynomials)	Multiple field extension generated by given algebraic elements represented by their minimal polynomials (not tested that they are irreducible)
static <E> MultivariateRing<MultivariatePolynomial<E>>	MultivariateRing(int nVariables, Ring<E> coefficientRing)	Ring of multivariate polynomials with specified number of variables over specified coefficient ring
static <E> MultivariateRing<MultivariatePolynomial<E>>	MultivariateRing(int nVariables, Ring<E> coefficientRing, Comparator<DegreeVector> monomialOrder)	Ring of multivariate polynomials with specified number of variables over specified coefficient ring
static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> MultivariateRing<Poly>	MultivariateRing(Poly factory)	Ring of multivariate polynomials with specified factory
static MultivariateRing<MultivariatePolynomial<Rational<BigInteger>>>	MultivariateRingQ(int nVariables)	Ring of multivariate polynomials over rationals (Q[x1, x2, ...])
static MultivariateRing<MultivariatePolynomial<BigInteger>>	MultivariateRingZ(int nVariables)	Ring of multivariate polynomials over integers (Z[x1, x2, ...])
static MultivariateRing<MultivariatePolynomial<BigInteger>>	MultivariateRingZp(int nVariables, BigInteger modulus)	Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...]) with arbitrary large modulus
static MultivariateRing<MultivariatePolynomialZp64>	MultivariateRingZp64(int nVariables, long modulus)	Ring of multivariate polynomials over Zp machine integers (Zp[x1, x2, ...])
static MultivariateRing<MultivariatePolynomialZp64>	MultivariateRingZp64(int nVariables, long modulus, Comparator<DegreeVector> monomialOrder)	Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...])
static MultivariateRing<MultivariatePolynomialZp64>	MultivariateRingZp64(int nVariables, IntegersZp64 modulus)	Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...])
static MultivariateRing<MultivariatePolynomialZp64>	MultivariateRingZp64(int nVariables, IntegersZp64 modulus, Comparator<DegreeVector> monomialOrder)	Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...])
static <Poly extends IPolynomial<Poly>> IPolynomialRing<Poly>	PolynomialRing(Poly factory)	Generic factory for polynomial ring
static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> QuotientRing<Term,Poly>	QuotientRing(MultivariateRing<Poly> baseRing, Ideal<Term,Poly> ideal)	Quotient ring baseRing/<ideal>
static <uPoly extends IUnivariatePolynomial<uPoly>> SimpleFieldExtension<uPoly>	SimpleFieldExtension(uPoly minimalPolynomial)	Returns a simple field extension generated by given minimal polynomial
static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly>	SplittingField(sPoly polynomial)	Splitting field of a given polynomial.
static <uPoly extends IUnivariatePolynomial<uPoly>> SimpleFieldExtension<uPoly>	UnivariateQuotientRing(uPoly modulus)	Deprecated. Use either GF(IUnivariatePolynomial) or AlgebraicNumberField(IUnivariatePolynomial)
static <E> UnivariateRing<UnivariatePolynomial<E>>	UnivariateRing(Ring<E> coefficientRing)	Ring of univariate polynomials over specified coefficient ring
static <Poly extends IUnivariatePolynomial<Poly>> UnivariateRing<Poly>	UnivariateRing(Poly factory)	Ring of univariate polynomials with specified factory
static UnivariateRing<UnivariatePolynomial<BigInteger>>	UnivariateRingZp(BigInteger modulus)	Ring of univariate polynomials over Zp integers (Zp[x]) with arbitrary large modulus
static UnivariateRing<UnivariatePolynomialZp64>	UnivariateRingZp64(long modulus)	Ring of univariate polynomials over Zp integers (Zp[x])
static UnivariateRing<UnivariatePolynomialZp64>	UnivariateRingZp64(IntegersZp64 modulus)	Ring of univariate polynomials over Zp integers (Zp[x])
static IntegersZp	Zp(long modulus)	Ring of integers modulo modulus (arbitrary large modulus)
static IntegersZp	Zp(BigInteger modulus)	Ring of integers modulo modulus (arbitrary large modulus)
static IntegersZp64	Zp64(long modulus)	Ring of integers modulo modulus (with modulus < 2^63)

Methods inherited from class java.lang.Object

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

Field Detail

Z

public static final Integers Z

Ring of integers (Z)

Q

public static final Rationals<BigInteger> Q

Field of rationals (Q)

GaussianRationals

public static AlgebraicNumberField<UnivariatePolynomial<Rational<BigInteger>>> GaussianRationals

Field of Gaussian rationals (rational complex numbers).

GaussianIntegers

public static AlgebraicNumberField<UnivariatePolynomial<BigInteger>> GaussianIntegers

Ring of Gaussian integers (integer complex numbers).

UnivariateRingZ

public static final UnivariateRing<UnivariatePolynomial<BigInteger>> UnivariateRingZ

Ring of univariate polynomials over integers (Z[x])

UnivariateRingQ

public static final UnivariateRing<UnivariatePolynomial<Rational<BigInteger>>> UnivariateRingQ

Ring of univariate polynomials over rationals (Q[x])

Method Detail

Frac

public static <E> Rationals<E> Frac(Ring<E> ring)

Ring of rational functions over specified ring

Parameters:

ring - the ring that numerators and denominators belong to

Zp64

public static IntegersZp64 Zp64(long modulus)

Ring of integers modulo modulus (with modulus < 2^63)

Parameters:

modulus - the modulus

Zp

public static IntegersZp Zp(long modulus)

Ring of integers modulo modulus (arbitrary large modulus)

Parameters:

modulus - the modulus (arbitrary large)

Zp

public static IntegersZp Zp(BigInteger modulus)

Ring of integers modulo modulus (arbitrary large modulus)

Parameters:

modulus - the modulus (arbitrary large)

GF

public static FiniteField<UnivariatePolynomialZp64> GF(long prime,
                                                       int exponent)

Galois field with the cardinality prime ^ exponent (with prime < 2^63).

Parameters:

prime - the integer prime modulus

exponent - the exponent (degree of modulo polynomial)

GF

public static FiniteField<UnivariatePolynomial<BigInteger>> GF(BigInteger prime,
                                                               int exponent)

Galois field with the cardinality prime ^ exponent for arbitrary large prime

Parameters:

prime - the integer (arbitrary large) prime modulus

exponent - the exponent (degree of modulo polynomial)

GF

public static <Poly extends IUnivariatePolynomial<Poly>> FiniteField<Poly> GF(Poly irreducible)

Galois field with the specified minimal polynomial. Note: there is no explicit check that minimal polynomial is irreducible

Parameters:

irreducible - irreducible univariate polynomial

AlgebraicNumberField

public static <Poly extends IUnivariatePolynomial<Poly>> AlgebraicNumberField<Poly> AlgebraicNumberField(Poly minimalPoly)

Algebraic number field generated by the specified minimal polynomial

GaussianNumbers

public static <E> AlgebraicNumberField<UnivariatePolynomial<E>> GaussianNumbers(Ring<E> ring)

Gaussian numbers for a given ring (that is ring adjoined with imaginary unit)

UnivariateQuotientRing

@Deprecated
public static <uPoly extends IUnivariatePolynomial<uPoly>> SimpleFieldExtension<uPoly> UnivariateQuotientRing(uPoly modulus)

Deprecated.

Use either GF(IUnivariatePolynomial) or AlgebraicNumberField(IUnivariatePolynomial)

Quotient ring baseRing/<modulus>

SimpleFieldExtension

public static <uPoly extends IUnivariatePolynomial<uPoly>> SimpleFieldExtension<uPoly> SimpleFieldExtension(uPoly minimalPolynomial)

Returns a simple field extension generated by given minimal polynomial

MultipleFieldExtension

public static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly> MultipleFieldExtension(sPoly... minimalPolynomials)

Multiple field extension generated by given algebraic elements represented by their minimal polynomials (not tested that they are irreducible)

SplittingField

public static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly> SplittingField(sPoly polynomial)

Splitting field of a given polynomial.

UnivariateRing

public static <E> UnivariateRing<UnivariatePolynomial<E>> UnivariateRing(Ring<E> coefficientRing)

Ring of univariate polynomials over specified coefficient ring

Parameters:

coefficientRing - the coefficient ring

UnivariateRing

public static <Poly extends IUnivariatePolynomial<Poly>> UnivariateRing<Poly> UnivariateRing(Poly factory)

Ring of univariate polynomials with specified factory

Parameters:

factory - factory

UnivariateRingZp64

public static UnivariateRing<UnivariatePolynomialZp64> UnivariateRingZp64(long modulus)

Ring of univariate polynomials over Zp integers (Zp[x])

Parameters:

modulus - the modulus

UnivariateRingZp64

public static UnivariateRing<UnivariatePolynomialZp64> UnivariateRingZp64(IntegersZp64 modulus)

Ring of univariate polynomials over Zp integers (Zp[x])

Parameters:

modulus - the modulus

UnivariateRingZp

public static UnivariateRing<UnivariatePolynomial<BigInteger>> UnivariateRingZp(BigInteger modulus)

Ring of univariate polynomials over Zp integers (Zp[x]) with arbitrary large modulus

Parameters:

modulus - the modulus (arbitrary large)

MultivariateRing

public static <E> MultivariateRing<MultivariatePolynomial<E>> MultivariateRing(int nVariables,
                                                                               Ring<E> coefficientRing,
                                                                               Comparator<DegreeVector> monomialOrder)

Ring of multivariate polynomials with specified number of variables over specified coefficient ring

Parameters:

nVariables - the number of variables

coefficientRing - the coefficient ring

monomialOrder - the monomial order

MultivariateRing

public static <E> MultivariateRing<MultivariatePolynomial<E>> MultivariateRing(int nVariables,
                                                                               Ring<E> coefficientRing)

Ring of multivariate polynomials with specified number of variables over specified coefficient ring

Parameters:

nVariables - the number of variables

coefficientRing - the coefficient ring

MultivariateRing

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> MultivariateRing<Poly> MultivariateRing(Poly factory)

Ring of multivariate polynomials with specified factory

Parameters:

factory - factory

MultivariateRingZ

public static MultivariateRing<MultivariatePolynomial<BigInteger>> MultivariateRingZ(int nVariables)

Ring of multivariate polynomials over integers (Z[x1, x2, ...])

Parameters:

nVariables - the number of variables

MultivariateRingQ

public static MultivariateRing<MultivariatePolynomial<Rational<BigInteger>>> MultivariateRingQ(int nVariables)

Ring of multivariate polynomials over rationals (Q[x1, x2, ...])

Parameters:

nVariables - the number of variables

MultivariateRingZp64

public static MultivariateRing<MultivariatePolynomialZp64> MultivariateRingZp64(int nVariables,
                                                                                long modulus,
                                                                                Comparator<DegreeVector> monomialOrder)

Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...])

Parameters:

nVariables - the number of variables

modulus - the modulus

monomialOrder - the monomial order

MultivariateRingZp64

public static MultivariateRing<MultivariatePolynomialZp64> MultivariateRingZp64(int nVariables,
                                                                                long modulus)

Ring of multivariate polynomials over Zp machine integers (Zp[x1, x2, ...])

Parameters:

nVariables - the number of variables

modulus - the modulus

MultivariateRingZp64

public static MultivariateRing<MultivariatePolynomialZp64> MultivariateRingZp64(int nVariables,
                                                                                IntegersZp64 modulus,
                                                                                Comparator<DegreeVector> monomialOrder)

Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...])

Parameters:

nVariables - the number of variables

modulus - the modulus

monomialOrder - monomial order

MultivariateRingZp64

public static MultivariateRing<MultivariatePolynomialZp64> MultivariateRingZp64(int nVariables,
                                                                                IntegersZp64 modulus)

Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...])

Parameters:

nVariables - the number of variables

modulus - the modulus

MultivariateRingZp

public static MultivariateRing<MultivariatePolynomial<BigInteger>> MultivariateRingZp(int nVariables,
                                                                                      BigInteger modulus)

Ring of multivariate polynomials over Zp integers (Zp[x1, x2, ...]) with arbitrary large modulus

Parameters:

nVariables - the number of variables

modulus - the modulus (arbitrary large)

PolynomialRing

public static <Poly extends IPolynomial<Poly>> IPolynomialRing<Poly> PolynomialRing(Poly factory)

Generic factory for polynomial ring

QuotientRing

public static <Term extends AMonomial<Term>,Poly extends AMultivariatePolynomial<Term,Poly>> QuotientRing<Term,Poly> QuotientRing(MultivariateRing<Poly> baseRing,
                                                                                                                                                    Ideal<Term,Poly> ideal)

Quotient ring baseRing/<ideal>