Inheritance diagram for nzmath.poly.ring.PolynomialRing:

Collaboration diagram for nzmath.poly.ring.PolynomialRing:

Public Member Functions
def	__init__ (self, coeffring, number_of_variables=1)

def	getCoefficientRing (self)

def	getQuotientField (self)

def	__eq__ (self, other)

def	__ne__ (self, other)

def	__hash__ (self)

def	__repr__ (self)

def	__str__ (self)

def	__contains__ (self, element)

def	issubring (self, other)

def	issuperring (self, other)

def	getCommonSuperring (self, other)

def	getCharacteristic (self)

def	createElement (self, seed)

def	gcd (self, a, b)

def	extgcd (self, a, b)

def	getInstance (cls, coeffring, number_of_variables=1)

Public Member Functions inherited from nzmath.ring.CommutativeRing
def	__init__ (self)

def	isdomain (self)

def	isnoetherian (self)

def	isufd (self)

def	ispid (self)

def	iseuclidean (self)

def	isfield (self)

def	registerModuleAction (self, action_ring, action)

def	hasaction (self, action_ring)

def	getaction (self, action_ring)

Public Attributes
	number_of_variables

Public Attributes inherited from nzmath.ring.CommutativeRing
	properties

Properties
	one = property(_get_one, None, None, "multiplicative unit")

	zero = property(_get_zero, None, None, "additive unit")

Private Member Functions
def	_zero_polynomial (self)

def	_constant_polynomial (self, seed)

def	_prepared_polynomial (self, preparation)

def	_get_one (self)

def	_get_zero (self)

Private Attributes
	_coefficient_ring

	_one

	_zero

Static Private Attributes
dictionary	_instances = {}

Detailed Description

The class of uni-/multivariate polynomial ring.
There's no need to specify the variable names.

Definition at line 12 of file ring.py.

Constructor & Destructor Documentation

◆ init()

def nzmath.poly.ring.PolynomialRing.__init__	(	self,
		coeffring,
		number_of_variables = `1`
	)

PolynomialRing(coeffring)
creates a polynomial ring for univariate polynomials, while
PolynomialRing(coeffring, n)
creates a polynomial ring for multivariate polynomials.

Definition at line 20 of file ring.py.

Member Function Documentation

◆ contains()

def nzmath.poly.ring.PolynomialRing.__contains__	(	self,
		element
	)

`in' operator is provided for checking the element be in the
ring.

Definition at line 93 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring.

◆ eq()

def nzmath.poly.ring.PolynomialRing.__eq__	(	self,
		other
	)

equality test

Reimplemented from nzmath.ring.Ring.

Definition at line 55 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

Referenced by nzmath.poly.ring.PolynomialRing.__ne__(), nzmath.quad.ReducedQuadraticForm.__ne__(), nzmath.ring.Ring.__ne__(), nzmath.real.RealField.__ne__(), nzmath.ring.Ideal.__ne__(), and nzmath.prime.FactoredInteger.__ne__().

◆ hash()

def nzmath.poly.ring.PolynomialRing.__hash__ ( self )

hash(self)

Reimplemented from nzmath.ring.Ring.

Definition at line 73 of file ring.py.

References nzmath.matrix.Matrix.__class__, nzmath.matrix.RingMatrix.__class__, nzmath.matrix.RingSquareMatrix.__class__, nzmath.matrix.FieldMatrix.__class__, nzmath.matrix.MatrixRing.__class__, nzmath.matrix.Subspace.__class__, nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

◆ ne()

def nzmath.poly.ring.PolynomialRing.__ne__	(	self,
		other
	)

not equal

Reimplemented from nzmath.ring.Ring.

Definition at line 67 of file ring.py.

References nzmath.group.Group.__eq__(), nzmath.poly.ratfunc.RationalFunction.__eq__(), nzmath.poly.multivar.TermIndices.__eq__(), nzmath.poly.ring.PolynomialRing.__eq__(), nzmath.poly.formalsum.FormalSumContainerInterface.__eq__(), nzmath.factor.misc.FactoredInteger.__eq__(), nzmath.imaginary.Complex.__eq__(), nzmath.poly.array.ArrayPoly.__eq__(), nzmath.group.GroupElement.__eq__(), nzmath.matrix.Matrix.__eq__(), nzmath.intresidue.IntegerResidueClass.__eq__(), nzmath.finitefield.FinitePrimeField.__eq__(), nzmath.imaginary.ComplexField.__eq__(), nzmath.poly.array.ArrayPolyMod.__eq__(), nzmath.module.Module.__eq__(), nzmath.permute.Permute.__eq__(), nzmath.intresidue.IntegerResidueClassRing.__eq__(), nzmath.poly.formalsum.DictFormalSum.__eq__(), nzmath.algfield.NumberField.__eq__(), nzmath.poly.formalsum.ListFormalSum.__eq__(), nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__eq__(), nzmath.finitefield.ExtendedFieldElement.__eq__(), nzmath.permute.ExPermute.__eq__(), nzmath.permute.PermGroup.__eq__(), nzmath.finitefield.ExtendedField.__eq__(), and nzmath.matrix.MatrixRing.__eq__().

◆ repr()

def nzmath.poly.ring.PolynomialRing.__repr__ ( self )

Return 'PolynomialRing(Ring, #vars)'

Definition at line 79 of file ring.py.

References nzmath.matrix.Matrix.__class__, nzmath.matrix.RingMatrix.__class__, nzmath.matrix.RingSquareMatrix.__class__, nzmath.matrix.FieldMatrix.__class__, nzmath.matrix.MatrixRing.__class__, nzmath.matrix.Subspace.__class__, nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

◆ str()

def nzmath.poly.ring.PolynomialRing.__str__ ( self )

Return R[][]

Definition at line 87 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

◆ _constant_polynomial()

def nzmath.poly.ring.PolynomialRing._constant_polynomial	(	self,
		seed
	)

private

Return a constant polynomial made from a constant seed.
seed should not be zero.

Definition at line 190 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

Referenced by nzmath.poly.ring.PolynomialRing.createElement().

◆ _get_one()

def nzmath.poly.ring.PolynomialRing._get_one ( self )

private

Definition at line 215 of file ring.py.

References nzmath.imaginary.ComplexField._one, nzmath.poly.ring.PolynomialRing._one, nzmath.algfield.NumberField._one, nzmath.intresidue.IntegerResidueClassRing._one, nzmath.finitefield.FinitePrimeField._one, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._one, nzmath.finitefield.ExtendedField._one, and nzmath.matrix.MatrixRing._one.

◆ _get_zero()

def nzmath.poly.ring.PolynomialRing._get_zero ( self )

private

Definition at line 223 of file ring.py.

References nzmath.imaginary.ComplexField._zero, nzmath.poly.ring.PolynomialRing._zero, nzmath.intresidue.IntegerResidueClassRing._zero, nzmath.algfield.NumberField._zero, nzmath.finitefield.FinitePrimeField._zero, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._zero, nzmath.finitefield.ExtendedField._zero, and nzmath.matrix.MatrixRing._zero.

◆ _prepared_polynomial()

def nzmath.poly.ring.PolynomialRing._prepared_polynomial	(	self,
		preparation
	)

private

Return a polynomial from given preparation, which is suited
for the first argument of uni-/multi-variable polynomials.

Definition at line 203 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

Referenced by nzmath.poly.ring.PolynomialRing.createElement().

◆ _zero_polynomial()

def nzmath.poly.ring.PolynomialRing._zero_polynomial ( self )

private

Return the zero polynomial in the polynomial ring.

Definition at line 179 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

Referenced by nzmath.poly.ring.PolynomialRing.createElement().

◆ createElement()

def nzmath.poly.ring.PolynomialRing.createElement	(	self,
		seed
	)

Return an element in the ring made from seed.

Reimplemented from nzmath.ring.Ring.

Definition at line 166 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing._constant_polynomial(), nzmath.poly.ring.PolynomialRing._prepared_polynomial(), and nzmath.poly.ring.PolynomialRing._zero_polynomial().

Referenced by nzmath.finitefield.FiniteField.random_element(), and nzmath.finitefield.FiniteField.TonelliShanks().

◆ extgcd()

def nzmath.poly.ring.PolynomialRing.extgcd	(	self,
		a,
		b
	)

Return the tuple (u, v, d): d is the greatest common divisor
of given polynomials, and they satisfy d = u*a + v*b. The
polynomials must be in the polynomial ring.  If the
coefficient ring is a field, the result is monic.

Definition at line 243 of file ring.py.

◆ gcd()

def nzmath.poly.ring.PolynomialRing.gcd	(	self,
		a,
		b
	)

Return the greatest common divisor of given polynomials.
The polynomials must be in the polynomial ring.
If the coefficient ring is a field, the result is monic.

Definition at line 231 of file ring.py.

Referenced by nzmath.rational.IntegerRing.lcm().

◆ getCharacteristic()

def nzmath.poly.ring.PolynomialRing.getCharacteristic ( self )

Return characteristic of the ring.

Reimplemented from nzmath.ring.Ring.

Definition at line 160 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring.

◆ getCoefficientRing()

def nzmath.poly.ring.PolynomialRing.getCoefficientRing ( self )

Return the coefficient ring.

Definition at line 41 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring.

Referenced by nzmath.poly.uniutil.DomainPolynomial.discriminant(), nzmath.poly.uniutil.FieldPolynomial.discriminant(), nzmath.poly.multiutil.PseudoDivisionProvider.exact_division(), nzmath.poly.multiutil.GcdProvider.gcd(), nzmath.poly.multiutil.PseudoDivisionProvider.pseudo_divmod(), nzmath.poly.multiutil.PseudoDivisionProvider.pseudo_floordiv(), nzmath.poly.multiutil.PseudoDivisionProvider.pseudo_mod(), nzmath.poly.uniutil.SubresultantGcdProvider.resultant(), nzmath.poly.uniutil.FieldPolynomial.resultant(), nzmath.poly.uniutil.SubresultantGcdProvider.subresultant_extgcd(), and nzmath.poly.uniutil.DomainPolynomial.to_field_polynomial().

◆ getCommonSuperring()

def nzmath.poly.ring.PolynomialRing.getCommonSuperring	(	self,
		other
	)

Return common superring of two rings.

Reimplemented from nzmath.ring.Ring.

Definition at line 140 of file ring.py.

◆ getInstance()

def nzmath.poly.ring.PolynomialRing.getInstance	(	cls,
		coeffring,
		number_of_variables = `1`
	)

Return an instance of the class with specified coefficient ring
and number of variables.

Definition at line 255 of file ring.py.

References nzmath.poly.ring.PolynomialRing._instances, nzmath.finitefield.FinitePrimeField._instances, nzmath.intresidue.IntegerResidueClassRing._instances, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._instances, and nzmath.matrix.MatrixRing._instances.

◆ getQuotientField()

def nzmath.poly.ring.PolynomialRing.getQuotientField ( self )

Return the quotient field of the ring if coefficient ring has
its quotient field.  Otherwise, an exception will be raised.

Reimplemented from nzmath.ring.CommutativeRing.

Definition at line 47 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

◆ issubring()

def nzmath.poly.ring.PolynomialRing.issubring	(	self,
		other
	)

reports whether another ring contains this polynomial ring.

Reimplemented from nzmath.ring.Ring.

Definition at line 105 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

Referenced by nzmath.ring.Ring.getCommonSuperring(), nzmath.rational.RationalField.getCommonSuperring(), and nzmath.rational.IntegerRing.getCommonSuperring().

◆ issuperring()

def nzmath.poly.ring.PolynomialRing.issuperring	(	self,
		other
	)

reports whether this polynomial ring contains another ring.

Reimplemented from nzmath.ring.Ring.

Definition at line 123 of file ring.py.

References nzmath.poly.ring.PolynomialRing._coefficient_ring, nzmath.poly.multiutil.RingElementProvider._coefficient_ring, nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring, nzmath.poly.ring.PolynomialRing.number_of_variables, nzmath.poly.ratfunc.RationalFunction.number_of_variables, nzmath.poly.multiutil.NestProvider.number_of_variables, nzmath.poly.multivar.BasicPolynomial.number_of_variables, and nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables.

Referenced by nzmath.ring.Ring.getCommonSuperring(), nzmath.poly.ring.PolynomialRing.getCommonSuperring(), nzmath.poly.ring.RationalFunctionField.getCommonSuperring(), nzmath.rational.RationalField.getCommonSuperring(), nzmath.rational.IntegerRing.getCommonSuperring(), nzmath.real.RealField.issubring(), and nzmath.poly.ring.RationalFunctionField.issuperring().

Member Data Documentation

◆ _coefficient_ring

nzmath.poly.ring.PolynomialRing._coefficient_ring

private

Definition at line 30 of file ring.py.

◆ _instances

dictionary nzmath.poly.ring.PolynomialRing._instances = {}

staticprivate

Definition at line 18 of file ring.py.

Referenced by nzmath.poly.ring.PolynomialRing.getInstance(), and nzmath.poly.ring.RationalFunctionField.getInstance().

◆ _one

nzmath.poly.ring.PolynomialRing._one

private

Definition at line 218 of file ring.py.

Referenced by nzmath.poly.ring.PolynomialRing._get_one(), nzmath.poly.ring.RationalFunctionField._get_one(), nzmath.real.RealField._getOne(), nzmath.ring.ResidueClassRing._getOne(), nzmath.rational.RationalField._getOne(), and nzmath.rational.IntegerRing._getOne().

◆ _zero

nzmath.poly.ring.PolynomialRing._zero

private

Definition at line 226 of file ring.py.

Referenced by nzmath.poly.ring.PolynomialRing._get_zero(), nzmath.poly.ring.RationalFunctionField._get_zero(), nzmath.real.RealField._getZero(), nzmath.ring.ResidueClassRing._getZero(), nzmath.rational.RationalField._getZero(), and nzmath.rational.IntegerRing._getZero().

◆ number_of_variables

nzmath.poly.ring.PolynomialRing.number_of_variables

Definition at line 39 of file ring.py.

Property Documentation

◆ one

nzmath.poly.ring.PolynomialRing.one = property(_get_one, None, None, "multiplicative unit")

static

Definition at line 221 of file ring.py.

Referenced by nzmath.ring.Field.gcd(), nzmath.finitefield.FiniteField.Legendre(), nzmath.finitefield.FiniteField.order(), nzmath.finitefield.FiniteField.sqrt(), and nzmath.finitefield.FiniteField.TonelliShanks().

◆ zero

nzmath.poly.ring.PolynomialRing.zero = property(_get_zero, None, None, "additive unit")

static

Definition at line 229 of file ring.py.

Referenced by nzmath.ring.Field.gcd().

The documentation for this class was generated from the following file:

nzmath/poly/ring.py

Public Member Functions

Public Attributes

Properties

Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ __init__()

Member Function Documentation

◆ __contains__()

◆ __eq__()

◆ __hash__()

◆ __ne__()

◆ __repr__()

◆ __str__()

◆ _constant_polynomial()

◆ _get_one()

◆ _get_zero()

◆ _prepared_polynomial()

◆ _zero_polynomial()

◆ createElement()

◆ extgcd()

◆ gcd()

◆ getCharacteristic()

◆ getCoefficientRing()

◆ getCommonSuperring()

◆ getInstance()

◆ getQuotientField()

◆ issubring()

◆ issuperring()

Member Data Documentation

◆ _coefficient_ring

◆ _instances

◆ _one

◆ _zero

◆ number_of_variables

Property Documentation

◆ one

◆ zero

◆ init()

◆ contains()

◆ eq()

◆ hash()

◆ ne()

◆ repr()

◆ str()