# UnivariateSkewPolynomialCategory RΒΆ

ore.spad line 46 [edit on github]

R: Ring

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
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)

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- 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: (%, %, %) -> %
from NonAssociativeRng

- binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing

- characteristic: () -> NonNegativeInteger
from NonAssociativeRing

- charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero

- coefficient: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

- coefficient: (%, NonNegativeInteger) -> R
from FreeModuleCategory(R, NonNegativeInteger)

- coefficient: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

- coefficients: % -> List R
from FreeModuleCategory(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
- coerce: Integer -> %
from NonAssociativeRing

- coerce: R -> %
from Algebra R

- commutator: (%, %) -> %
from NonAssociativeRng

- construct: List Record(k: NonNegativeInteger, c: R) -> %
from IndexedProductCategory(R, NonNegativeInteger)

- constructOrdered: List Record(k: NonNegativeInteger, c: R) -> %
from IndexedProductCategory(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, %) -> %

- ground: % -> R

- hash: % -> SingleInteger
from SetCategory

- hashUpdate!: (HashState, %) -> HashState
from SetCategory

- latex: % -> String
from SetCategory

- leadingCoefficient: % -> R
from IndexedProductCategory(R, NonNegativeInteger)

- leadingMonomial: % -> %
from IndexedProductCategory(R, NonNegativeInteger)

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

- linearExtend: (NonNegativeInteger -> R, %) -> R if R has CommutativeRing
from FreeModuleCategory(R, NonNegativeInteger)

- listOfTerms: % -> List Record(k: NonNegativeInteger, c: R)

- mainVariable: % -> Union(SingletonAsOrderedSet, failed)
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

- map: (R -> R, %) -> %
from IndexedProductCategory(R, NonNegativeInteger)

- mapExponents: (NonNegativeInteger -> 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
from IndexedProductCategory(R, NonNegativeInteger)

- monomial: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

- monomial: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

- monomial: (R, NonNegativeInteger) -> %
from IndexedProductCategory(R, NonNegativeInteger)

- monomials: % -> List %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

- one?: % -> Boolean
from MagmaWithUnit

- opposite?: (%, %) -> Boolean
from AbelianMonoid

- pomopo!: (%, R, NonNegativeInteger, %) -> %

- primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

- 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: % -> %
from IndexedProductCategory(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)

- support: % -> List NonNegativeInteger
from FreeModuleCategory(R, NonNegativeInteger)

- 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

AbelianMonoidRing(R, NonNegativeInteger)

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

BiModule(%, %)

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

BiModule(R, R)

canonicalUnitNormal if R has canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

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)

FreeModuleCategory(R, NonNegativeInteger)

IndexedDirectProductCategory(R, NonNegativeInteger)

IndexedProductCategory(R, NonNegativeInteger)

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

LeftModule Fraction Integer if R has Algebra Fraction Integer

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

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

noZeroDivisors if R has EntireRing

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RightModule Fraction Integer if R has Algebra Fraction Integer

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