UnivariateSkewPolynomial(x, R, sigma, delta)ΒΆ

ore.spad line 516

This is the domain of univariate skew polynomials over an Ore coefficient field in a named variable. The multiplication is given by x a = \sigma(a) x + \delta a.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> % if R has Algebra Fraction Integer
from RightModule Fraction Integer
*: (%, R) -> %
from RightModule R
*: (Fraction Integer, %) -> % if R has Algebra Fraction Integer
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, R) -> % if R has Field
from AbelianMonoidRing(R, NonNegativeInteger)
=: (%, %) -> Boolean
from BasicType
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
apply: (%, R, R) -> R
from UnivariateSkewPolynomialCategory R
associates?: (%, %) -> Boolean if R has EntireRing
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
from CharacteristicNonZero
coefficient: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
coefficient: (%, NonNegativeInteger) -> R
from AbelianMonoidRing(R, NonNegativeInteger)
coefficient: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
coefficients: % -> List R
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
coerce: % -> % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has Algebra Fraction Integer or R has RetractableTo Fraction Integer
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: R -> %
from Algebra R
coerce: Variable x -> %
coerce(x) returns x as a skew-polynomial.
commutator: (%, %) -> %
from NonAssociativeRng
content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
degree: % -> NonNegativeInteger
from AbelianMonoidRing(R, NonNegativeInteger)
degree: (%, List SingletonAsOrderedSet) -> List NonNegativeInteger
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
degree: (%, SingletonAsOrderedSet) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
exquo: (%, %) -> Union(%, failed) if R has EntireRing
from EntireRing
exquo: (%, R) -> Union(%, failed) if R has EntireRing
from UnivariateSkewPolynomialCategory R
fmecg: (%, NonNegativeInteger, R, %) -> %
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
ground: % -> R
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
ground?: % -> Boolean
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> R
from AbelianMonoidRing(R, NonNegativeInteger)
leadingMonomial: % -> %
from AbelianMonoidRing(R, NonNegativeInteger)
leftDivide: (%, %) -> Record(quotient: %, remainder: %) if R has Field
from UnivariateSkewPolynomialCategory R
leftExactQuotient: (%, %) -> Union(%, failed) if R has Field
from UnivariateSkewPolynomialCategory R
leftExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field
from UnivariateSkewPolynomialCategory R
leftGcd: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
leftLcm: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftQuotient: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
leftRemainder: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
mainVariable: % -> Union(SingletonAsOrderedSet, failed)
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
map: (R -> R, %) -> %
from AbelianMonoidRing(R, NonNegativeInteger)
mapExponents: (NonNegativeInteger -> NonNegativeInteger, %) -> %
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
minimumDegree: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
monicLeftDivide: (%, %) -> Record(quotient: %, remainder: %) if R has IntegralDomain
from UnivariateSkewPolynomialCategory R
monicRightDivide: (%, %) -> Record(quotient: %, remainder: %) if R has IntegralDomain
from UnivariateSkewPolynomialCategory R
monomial: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
monomial: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
monomial: (R, NonNegativeInteger) -> %
from AbelianMonoidRing(R, NonNegativeInteger)
monomial?: % -> Boolean
from AbelianMonoidRing(R, NonNegativeInteger)
monomials: % -> List %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
pomopo!: (%, R, NonNegativeInteger, %) -> %
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
primitivePart: % -> % if R has GcdDomain
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
recip: % -> Union(%, failed)
from MagmaWithUnit
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)
from LinearlyExplicitOver R
reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix R
from LinearlyExplicitOver R
reductum: % -> %
from AbelianMonoidRing(R, NonNegativeInteger)
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer
retract: % -> R
from RetractableTo R
retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(R, failed)
from RetractableTo R
right_ext_ext_GCD: (%, %) -> Record(generator: %, coef1: %, coef2: %, coefu: %, coefv: %) if R has Field
from UnivariateSkewPolynomialCategory R
rightDivide: (%, %) -> Record(quotient: %, remainder: %) if R has Field
from UnivariateSkewPolynomialCategory R
rightExactQuotient: (%, %) -> Union(%, failed) if R has Field
from UnivariateSkewPolynomialCategory R
rightExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field
from UnivariateSkewPolynomialCategory R
rightGcd: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
rightLcm: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightQuotient: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
rightRemainder: (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory R
sample: %
from AbelianMonoid
smaller?: (%, %) -> Boolean if R has Comparable
from Comparable
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
totalDegree: % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
totalDegree: (%, List SingletonAsOrderedSet) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
totalDegreeSorted: (%, List SingletonAsOrderedSet) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
unit?: % -> Boolean if R has EntireRing
from EntireRing
unitCanonical: % -> % if R has EntireRing
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has EntireRing
from EntireRing
variables: % -> List SingletonAsOrderedSet
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, NonNegativeInteger)

AbelianSemiGroup

Algebra % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if R has Algebra Fraction Integer

BiModule(R, R)

CancellationAbelianMonoid

canonicalUnitNormal if R has canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

CommutativeStar if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Comparable if R has Comparable

EntireRing if R has EntireRing

FiniteAbelianMonoidRing(R, NonNegativeInteger)

FullyLinearlyExplicitOver R

FullyRetractableTo R

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

Module % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RightModule %

RightModule Fraction Integer if R has Algebra Fraction Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown

UnivariateSkewPolynomialCategory R