SparseMultivariateSkewPolynomial(R, Var, sigma, delta)

skpol.spad line 250 [edit on github]

• R : Ring
• Var : OrderedSet
• sigma : Mapping(Automorphism(R), Var)
• delta : Mapping(Mapping(R, R), Var)

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.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

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

from RightModule(Fraction(Integer))

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

from RightModule(Integer)

* : (R, %) -> %

from LeftModule(R)

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

from LeftModule(Fraction(Integer))

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

/ : (%, R) -> % if R has Field

from AbelianMonoidRing(R, IndexedExponents(Var))

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

D : Var -> %

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

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

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

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

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 : (%, Var, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)

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

from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)

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

from AbelianMonoidRing(R, IndexedExponents(Var))

coefficients : % -> List(R)

from FreeModuleCategory(R, IndexedExponents(Var))

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

from Algebra(%)

coerce : R -> %

from Algebra(R)

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

from Algebra(Fraction(Integer))

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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))

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)

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 : % -> R

from FiniteAbelianMonoidRing(R, IndexedExponents(Var))

ground? : % -> Boolean

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

from IndexedProductCategory(R, IndexedExponents(Var))

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

from FiniteAbelianMonoidRing(R, IndexedExponents(Var))

minimumDegree : % -> IndexedExponents(Var)

from FiniteAbelianMonoidRing(R, IndexedExponents(Var))

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

from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)

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

from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)

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

from IndexedProductCategory(R, IndexedExponents(Var))

monomial? : % -> Boolean

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 R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has Algebra(Fraction(Integer)) or R has CommutativeRing and % has VariablesCommuteWithCoefficients

from NonAssociativeAlgebra(%)

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(%) -> Matrix(R)

from LinearlyExplicitOver(R)

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

from LinearlyExplicitOver(Integer)

reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(R), vec : Vector(R))

from LinearlyExplicitOver(R)

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

from LinearlyExplicitOver(Integer)

reductum : % -> %

from IndexedProductCategory(R, IndexedExponents(Var))

retract : % -> R

from RetractableTo(R)

retract : % -> Fraction(Integer) if R has RetractableTo(Fraction(Integer))

from RetractableTo(Fraction(Integer))

retract : % -> Integer if R has RetractableTo(Integer)

from RetractableTo(Integer)

retractIfCan : % -> Union(R, "failed")

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)

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

~= : (%, %) -> Boolean

from BasicType

Module(Fraction(Integer))

NonAssociativeSemiRing

LeftModule(R)

BiModule(%, %)

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

FullyRetractableTo(R)

SemiRing

EntireRing

MultivariateSkewPolynomialCategory(R, IndexedExponents(Var), Var)

NonAssociativeAlgebra(Fraction(Integer))

FreeModuleCategory(R, IndexedExponents(Var))

unitsKnown

FullyLinearlyExplicitOver(R)

FiniteAbelianMonoidRing(R, IndexedExponents(Var))

CharacteristicNonZero

IndexedProductCategory(R, IndexedExponents(Var))

noZeroDivisors

Magma

SemiGroup

RightModule(R)

IntegralDomain

LeftModule(%)

NonAssociativeRing

CharacteristicZero

MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)

Module(R)

CommutativeRing

Algebra(%)

BiModule(R, R)

Algebra(R)

LinearlyExplicitOver(R)

NonAssociativeSemiRng

CancellationAbelianMonoid

Comparable

RetractableTo(Integer)

RightModule(Fraction(Integer))

AbelianMonoidRing(R, IndexedExponents(Var))

CommutativeStar

AbelianMonoid

MagmaWithUnit

RightModule(%)

AbelianProductCategory(R)

Module(%)

CoercibleTo(OutputForm)

SemiRng

LinearlyExplicitOver(Integer)

Monoid

NonAssociativeAlgebra(R)

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(Integer)

LeftModule(Fraction(Integer))

IndexedDirectProductCategory(R, IndexedExponents(Var))

SetCategory

CoercibleFrom(Fraction(Integer))

NonAssociativeRng

CoercibleFrom(R)

BiModule(Fraction(Integer), Fraction(Integer))

RetractableTo(Fraction(Integer))

RetractableTo(R)

AbelianGroup

AbelianSemiGroup