UnivariateSkewPolynomialCategory RΒΆ

ore.spad line 46

This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by x a = \sigma(a) x + \delta a. This category is an evolution of the types MonogenicLinearOperator, OppositeMonogenicLinearOperator, and NonCommutativeOperatorDivision developped by Jean Della Dora and Stephen M. Watt.

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
apply(p, c, m) returns p(m) where the action is given by x m = c sigma(m) + delta(m).
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 IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer or R has Algebra Fraction Integer
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: R -> %
from Algebra R
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
exquo(l, a) returns the exact quotient of l by a, returning "failed" if this is not possible.
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
leftDivide(a, b) returns the pair [q, r] such that a = b*q + r and the degree of r is less than the degree of b. This process is called left division\ ``''.
leftExactQuotient: (%, %) -> Union(%, failed) if R has Field
leftExactQuotient(a, b) computes the value q, if it exists, such that a = b*q.
leftExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field
leftExtendedGcd(a, b) returns [c, d, g] such that g = a * c + b * d = leftGcd(a, b).
leftGcd: (%, %) -> % if R has Field
leftGcd(a, b) computes the value g of highest degree such that a = g*aa b = g*bb for some values aa and bb. The value g is computed using left-division.
leftLcm: (%, %) -> % if R has Field
leftLcm(a, b) computes the value m of lowest degree such that m = aa*a = bb*b for some values aa and bb. The value m is computed using right-division.
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftQuotient: (%, %) -> % if R has Field
leftQuotient(a, b) computes the pair [q, r] such that a = b*q + r and the degree of r is less than the degree of b. The value q is returned.
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
leftRemainder: (%, %) -> % if R has Field
leftRemainder(a, b) computes the pair [q, r] such that a = b*q + r and the degree of r is less than the degree of b. The value r is returned.
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
monicLeftDivide(a, b) returns the pair [q, r] such that a = b*q + r and the degree of r is less than the degree of b. b must be monic. This process is called left division\ ``''.
monicRightDivide: (%, %) -> Record(quotient: %, remainder: %) if R has IntegralDomain
monicRightDivide(a, b) returns the pair [q, r] such that a = q*b + r and the degree of r is less than the degree of b. b must be monic. This process is called right division\ ``''.
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
right_ext_ext_GCD(a, b) returns g, c, d, u, v such that g = c * a + d * b = rightGcd(a, b), u * a = - v * b = leftLcm(a, b) and matrix matrix([[c, d], [u, v]]) is invertible.
rightDivide: (%, %) -> Record(quotient: %, remainder: %) if R has Field
rightDivide(a, b) returns the pair [q, r] such that a = q*b + r and the degree of r is less than the degree of b. This process is called right division\ ``''.
rightExactQuotient: (%, %) -> Union(%, failed) if R has Field
rightExactQuotient(a, b) computes the value q, if it exists such that a = q*b.
rightExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field
rightExtendedGcd(a, b) returns [c, d, g] such that g = c * a + d * b = rightGcd(a, b).
rightGcd: (%, %) -> % if R has Field
rightGcd(a, b) computes the value g of highest degree such that a = aa*g b = bb*g for some values aa and bb. The value g is computed using right-division.
rightLcm: (%, %) -> % if R has Field
rightLcm(a, b) computes the value m of lowest degree such that m = a*aa = b*bb for some values aa and bb. The value m is computed using left-division.
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightQuotient: (%, %) -> % if R has Field
rightQuotient(a, b) computes the pair [q, r] such that a = q*b + r and the degree of r is less than the degree of b. The value q is returned.
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
rightRemainder: (%, %) -> % if R has Field
rightRemainder(a, b) computes the pair [q, r] such that a = q*b + r and the degree of r is less than the degree of b. The value r is returned.
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 IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing 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 CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

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