SparseMultivariateSkewPolynomial(R, Var, sigma, delta)ΒΆ

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SparseMultivariateSkewPolynomial(R, Var, sigma, delta) defines a mutivariate Ore ring over R in variables from V. sigma(v) gives automorphism of R corresponding to variable v and delta(v) gives corresponding derivative.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

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

from RightModule Fraction Integer

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

from RightModule 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, IndexedExponents Var)

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean if R has EntireRing

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

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

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

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

from CharacteristicNonZero

coefficient: (%, IndexedExponents Var) -> R

from AbelianMonoidRing(R, IndexedExponents Var)

coefficient: (%, List Var, List NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

coefficient: (%, Var, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

coefficients: % -> List R

from FreeModuleCategory(R, IndexedExponents Var)

coerce: % -> % if % has VariablesCommuteWithCoefficients and R has CommutativeRing 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

commutator: (%, %) -> %

from NonAssociativeRng

construct: List Record(k: IndexedExponents Var, c: R) -> %

from IndexedProductCategory(R, IndexedExponents Var)

constructOrdered: List Record(k: IndexedExponents Var, c: R) -> %

from IndexedProductCategory(R, IndexedExponents Var)

content: % -> R if R has GcdDomain

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

D: Var -> %

D(v) returns operator corresponding to derivative with respect to v in R.

degree: % -> IndexedExponents Var

from AbelianMonoidRing(R, IndexedExponents Var)

degree: (%, List Var) -> List NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

degree: (%, Var) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

Delta: Symbol -> % if Var has variable: Symbol -> Var

Delta(s) returns operator corresponding to derivative with respect to s in R.

exquo: (%, %) -> Union(%, failed) if R has EntireRing

from EntireRing

exquo: (%, R) -> Union(%, failed) if R has EntireRing

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

fmecg: (%, IndexedExponents Var, R, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

ground?: % -> Boolean

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

ground: % -> R

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

latex: % -> String

from SetCategory

leadingCoefficient: % -> R

from IndexedProductCategory(R, IndexedExponents Var)

leadingMonomial: % -> %

from IndexedProductCategory(R, IndexedExponents Var)

leadingSupport: % -> IndexedExponents Var

from IndexedProductCategory(R, IndexedExponents Var)

leadingTerm: % -> Record(k: IndexedExponents Var, c: R)

from IndexedProductCategory(R, IndexedExponents Var)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

linearExtend: (IndexedExponents Var -> R, %) -> R if R has CommutativeRing

from FreeModuleCategory(R, IndexedExponents Var)

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

from IndexedDirectProductCategory(R, IndexedExponents Var)

mainVariable: % -> Union(Var, failed)

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

map: (R -> R, %) -> %

from IndexedProductCategory(R, IndexedExponents Var)

mapExponents: (IndexedExponents Var -> IndexedExponents Var, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

minimumDegree: % -> IndexedExponents Var

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

monomial?: % -> Boolean

from IndexedProductCategory(R, IndexedExponents Var)

monomial: (%, List Var, List NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

monomial: (%, Var, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

monomial: (R, IndexedExponents Var) -> %

from IndexedProductCategory(R, IndexedExponents Var)

monomials: % -> List %

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(R, IndexedExponents Var)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> % if % has VariablesCommuteWithCoefficients and R has CommutativeRing or R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has Algebra Fraction Integer

from NonAssociativeAlgebra Fraction Integer

pomopo!: (%, R, IndexedExponents Var, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

primitiveMonomials: % -> List %

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

primitivePart: % -> % if R has GcdDomain

from FiniteAbelianMonoidRing(R, IndexedExponents Var)

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 IndexedProductCategory(R, IndexedExponents Var)

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

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

smaller?: (%, %) -> Boolean if R has Comparable

from Comparable

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

support: % -> List IndexedExponents Var

from FreeModuleCategory(R, IndexedExponents Var)

totalDegree: % -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

totalDegree: (%, List Var) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

totalDegreeSorted: (%, List Var) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

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 Var

from MaybeSkewPolynomialCategory(R, IndexedExponents Var, Var)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, IndexedExponents Var)

AbelianProductCategory R

AbelianSemiGroup

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

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if % has VariablesCommuteWithCoefficients and 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

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom R

CoercibleTo OutputForm

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

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

Comparable if R has Comparable

EntireRing if R has EntireRing

FiniteAbelianMonoidRing(R, IndexedExponents Var)

FreeModuleCategory(R, IndexedExponents Var)

FullyLinearlyExplicitOver R

FullyRetractableTo R

IndexedDirectProductCategory(R, IndexedExponents Var)

IndexedProductCategory(R, IndexedExponents Var)

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, IndexedExponents Var, Var)

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

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid

MultivariateSkewPolynomialCategory(R, IndexedExponents Var, Var)

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

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

NonAssociativeAlgebra R if % has VariablesCommuteWithCoefficients and R has CommutativeRing

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 Integer if R has LinearlyExplicitOver Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

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

unitsKnown