# MultivariateSkewPolynomialCategory(R, E, Var)ΒΆ

- R: Ring
- E: OrderedAbelianMonoidSup
- Var: OrderedSet

undocumented

- 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, E)
- =: (%, %) -> 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, E)
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
- from CharacteristicNonZero
- coefficient: (%, E) -> R
- from AbelianMonoidRing(R, E)
- coefficient: (%, List Var, List NonNegativeInteger) -> %
- from MaybeSkewPolynomialCategory(R, E, Var)
- coefficient: (%, Var, NonNegativeInteger) -> %
- from MaybeSkewPolynomialCategory(R, E, Var)
- coefficients: % -> List R
- from FiniteAbelianMonoidRing(R, E)
- 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, E)
- degree: % -> E
- from AbelianMonoidRing(R, E)
- degree: (%, List Var) -> List NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, Var)
- degree: (%, Var) -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, Var)
- exquo: (%, %) -> Union(%, failed) if R has EntireRing
- from EntireRing
- exquo: (%, R) -> Union(%, failed) if R has EntireRing
- from FiniteAbelianMonoidRing(R, E)
- fmecg: (%, E, R, %) -> %
- from FiniteAbelianMonoidRing(R, E)
- ground: % -> R
- from FiniteAbelianMonoidRing(R, E)
- ground?: % -> Boolean
- from FiniteAbelianMonoidRing(R, E)
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- latex: % -> String
- from SetCategory
- leadingCoefficient: % -> R
- from AbelianMonoidRing(R, E)
- leadingMonomial: % -> %
- from AbelianMonoidRing(R, E)
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- mainVariable: % -> Union(Var, failed)
- from MaybeSkewPolynomialCategory(R, E, Var)
- map: (R -> R, %) -> %
- from AbelianMonoidRing(R, E)
- mapExponents: (E -> E, %) -> %
- from FiniteAbelianMonoidRing(R, E)
- minimumDegree: % -> E
- from FiniteAbelianMonoidRing(R, E)
- monomial: (%, List Var, List NonNegativeInteger) -> %
- from MaybeSkewPolynomialCategory(R, E, Var)
- monomial: (%, Var, NonNegativeInteger) -> %
- from MaybeSkewPolynomialCategory(R, E, Var)
- monomial: (R, E) -> %
- from AbelianMonoidRing(R, E)
- monomial?: % -> Boolean
- from AbelianMonoidRing(R, E)
- monomials: % -> List %
- from MaybeSkewPolynomialCategory(R, E, Var)
- numberOfMonomials: % -> NonNegativeInteger
- from FiniteAbelianMonoidRing(R, E)
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- pomopo!: (%, R, E, %) -> %
- from FiniteAbelianMonoidRing(R, E)
- primitiveMonomials: % -> List %
- from MaybeSkewPolynomialCategory(R, E, Var)
- primitivePart: % -> % if R has GcdDomain
- from FiniteAbelianMonoidRing(R, E)
- 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, E)
- 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
- totalDegree: % -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, Var)
- totalDegree: (%, List Var) -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, Var)
- totalDegreeSorted: (%, List Var) -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, 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, E, Var)
- zero?: % -> Boolean
- from AbelianMonoid

AbelianMonoidRing(R, E)

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 and % has VariablesCommuteWithCoefficients

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

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, E)

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, E, Var)

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 and % has VariablesCommuteWithCoefficients

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