UnivariateSkewPolynomialCategory R¶

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 developed by Jean Della Dora and Stephen M. Watt.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> % if R has Algebra Fraction Integer
*: (%, R) -> %

from RightModule R

*: (Fraction Integer, %) -> % if R has Algebra 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
=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
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: (%, %, %) -> %
binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
coefficient: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
coefficient: (%, NonNegativeInteger) -> R
coefficient: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
coefficients: % -> List R
coerce: % -> % if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

from Algebra %

coerce: % -> OutputForm
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer or R has Algebra Fraction Integer
coerce: Integer -> %
coerce: R -> %

from Algebra R

commutator: (%, %) -> %
construct: List Record(k: NonNegativeInteger, c: R) -> %
constructOrdered: List Record(k: NonNegativeInteger, c: R) -> %
content: % -> R if R has GcdDomain
degree: % -> NonNegativeInteger
degree: (%, List SingletonAsOrderedSet) -> List NonNegativeInteger
degree: (%, SingletonAsOrderedSet) -> NonNegativeInteger
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, %) -> %
ground?: % -> Boolean
ground: % -> R
hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

latex: % -> String

from SetCategory

leadingTerm: % -> Record(k: NonNegativeInteger, c: R)
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.

linearExtend: (NonNegativeInteger -> R, %) -> R if R has CommutativeRing
listOfTerms: % -> List Record(k: NonNegativeInteger, c: R)
mainVariable: % -> Union(SingletonAsOrderedSet, failed)
map: (R -> R, %) -> %
mapExponents: (NonNegativeInteger -> NonNegativeInteger, %) -> %
minimumDegree: % -> 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?: % -> Boolean
monomial: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
monomial: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
monomial: (R, NonNegativeInteger) -> %
monomials: % -> List %
numberOfMonomials: % -> NonNegativeInteger
one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

pomopo!: (%, R, NonNegativeInteger, %) -> %
primitiveMonomials: % -> List %
primitivePart: % -> % if R has GcdDomain
recip: % -> Union(%, failed)

from MagmaWithUnit

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)

from LinearlyExplicitOver R

reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix R

from LinearlyExplicitOver R

reductum: % -> %
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
retract: % -> R

from RetractableTo R

retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has 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)
support: % -> List NonNegativeInteger
totalDegree: % -> NonNegativeInteger
totalDegree: (%, List SingletonAsOrderedSet) -> NonNegativeInteger
totalDegreeSorted: (%, List SingletonAsOrderedSet) -> NonNegativeInteger
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
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

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

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

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

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

Magma

MagmaWithUnit

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

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

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

unitsKnown