SparseUnivariateSkewPolynomial(R, sigma, delta)

ore.spad line 480 [edit on github]

This is the domain of sparse univariate skew polynomials over an Ore coefficient field. The multiplication is given by x a = \sigma(a) x + \delta a.

* : (%, %) -> %
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, NonNegativeInteger)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
apply : (%, R, R) -> R
from UnivariateSkewPolynomialCategory(R)
associates? : (%, %) -> Boolean if R has EntireRing
from EntireRing
associator : (%, %, %) -> %
from NonAssociativeRng
binomThmExpt : (%, %, NonNegativeInteger) -> % if % has CommutativeRing
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
charthRoot : % -> Union(%, "failed") if R has CharacteristicNonZero
from CharacteristicNonZero
coefficient : (%, List(SingletonAsOrderedSet), List(NonNegativeInteger)) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
coefficient : (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
coefficient : (%, NonNegativeInteger) -> R
from FreeModuleCategory(R, NonNegativeInteger)
coefficients : % -> List(R)
from FreeModuleCategory(R, NonNegativeInteger)
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 : NonNegativeInteger, c : R)) -> %
from IndexedProductCategory(R, NonNegativeInteger)
constructOrdered : List(Record(k : NonNegativeInteger, c : R)) -> %
from IndexedProductCategory(R, NonNegativeInteger)
content : % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
degree : (%, List(SingletonAsOrderedSet)) -> List(NonNegativeInteger)
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
degree : % -> NonNegativeInteger
from AbelianMonoidRing(R, NonNegativeInteger)
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
from UnivariateSkewPolynomialCategory(R)
fmecg : (%, NonNegativeInteger, R, %) -> %
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
ground : % -> R
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
ground? : % -> Boolean
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
latex : % -> String
from SetCategory
leadingCoefficient : % -> R
from IndexedProductCategory(R, NonNegativeInteger)
leadingMonomial : % -> %
from IndexedProductCategory(R, NonNegativeInteger)
leadingSupport : % -> NonNegativeInteger
from IndexedProductCategory(R, NonNegativeInteger)
leadingTerm : % -> Record(k : NonNegativeInteger, c : R)
from IndexedProductCategory(R, NonNegativeInteger)
leftDivide : (%, %) -> Record(quotient : %, remainder : %) if R has Field
from UnivariateSkewPolynomialCategory(R)
leftExactQuotient : (%, %) -> Union(%, "failed") if R has Field
from UnivariateSkewPolynomialCategory(R)
leftExtendedGcd : (%, %) -> Record(coef1 : %, coef2 : %, generator : %) if R has Field
from UnivariateSkewPolynomialCategory(R)
leftGcd : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
leftLcm : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
leftPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower : (%, PositiveInteger) -> %
from Magma
leftQuotient : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
leftRecip : % -> Union(%, "failed")
from MagmaWithUnit
leftRemainder : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
linearExtend : (Mapping(R, NonNegativeInteger), %) -> R if R has CommutativeRing
from FreeModuleCategory(R, NonNegativeInteger)
listOfTerms : % -> List(Record(k : NonNegativeInteger, c : R))
from IndexedDirectProductCategory(R, NonNegativeInteger)
mainVariable : % -> Union(SingletonAsOrderedSet, "failed")
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
map : (Mapping(R, R), %) -> %
from IndexedProductCategory(R, NonNegativeInteger)
mapExponents : (Mapping(NonNegativeInteger, NonNegativeInteger), %) -> %
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
minimumDegree : % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
monicLeftDivide : (%, %) -> Record(quotient : %, remainder : %) if R has IntegralDomain
from UnivariateSkewPolynomialCategory(R)
monicRightDivide : (%, %) -> Record(quotient : %, remainder : %) if R has IntegralDomain
from UnivariateSkewPolynomialCategory(R)
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)
monomial? : % -> Boolean
from IndexedProductCategory(R, NonNegativeInteger)
monomials : % -> List(%)
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
numberOfMonomials : % -> NonNegativeInteger
from IndexedDirectProductCategory(R, NonNegativeInteger)
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
outputForm : (%, OutputForm) -> OutputForm

outputForm(p, x) returns the output form of p using x for the otherwise anonymous variable.

plenaryPower : (%, PositiveInteger) -> % if R has Algebra(Fraction(Integer)) or R has CommutativeRing
from NonAssociativeAlgebra(R)
pomopo! : (%, R, NonNegativeInteger, %) -> %
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
primitiveMonomials : % -> List(%)
from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)
primitivePart : % -> % if R has GcdDomain
from FiniteAbelianMonoidRing(R, NonNegativeInteger)
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, NonNegativeInteger)
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)
rightDivide : (%, %) -> Record(quotient : %, remainder : %) if R has Field
from UnivariateSkewPolynomialCategory(R)
rightExactQuotient : (%, %) -> Union(%, "failed") if R has Field
from UnivariateSkewPolynomialCategory(R)
rightExtendedGcd : (%, %) -> Record(coef1 : %, coef2 : %, generator : %) if R has Field
from UnivariateSkewPolynomialCategory(R)
rightGcd : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
rightLcm : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightQuotient : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
rightRemainder : (%, %) -> % if R has Field
from UnivariateSkewPolynomialCategory(R)
right_ext_ext_GCD : (%, %) -> Record(generator : %, coef1 : %, coef2 : %, coefu : %, coefv : %) if R has Field
from UnivariateSkewPolynomialCategory(R)
sample : () -> %
from AbelianMonoid
smaller? : (%, %) -> Boolean if R has Comparable
from Comparable
subtractIfCan : (%, %) -> Union(%, "failed")
from CancellationAbelianMonoid
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
~= : (%, %) -> Boolean
from BasicType

CharacteristicNonZero

Module(Fraction(Integer))

NonAssociativeSemiRing

LeftModule(R)

BiModule(%, %)

FreeModuleCategory(R, NonNegativeInteger)

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

FullyRetractableTo(R)

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

FullyLinearlyExplicitOver(R)

CoercibleTo(OutputForm)

noZeroDivisors

Magma

SemiGroup

IntegralDomain

LeftModule(%)

IndexedProductCategory(R, NonNegativeInteger)

NonAssociativeRing

CharacteristicZero

Module(R)

CommutativeRing

Algebra(%)

UnivariateSkewPolynomialCategory(R)

BiModule(R, R)

RightModule(Fraction(Integer))

Algebra(R)

RightModule(R)

FiniteAbelianMonoidRing(R, NonNegativeInteger)

NonAssociativeSemiRng

CancellationAbelianMonoid

Comparable

RetractableTo(Integer)

CommutativeStar

AbelianMonoid

MagmaWithUnit

RightModule(%)

AbelianProductCategory(R)

Module(%)

LinearlyExplicitOver(Integer)

SemiRng

Monoid

IndexedDirectProductCategory(R, NonNegativeInteger)

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(Integer)

AbelianSemiGroup

SetCategory

CoercibleFrom(Fraction(Integer))

LinearlyExplicitOver(R)

LeftModule(Fraction(Integer))

NonAssociativeRng

AbelianMonoidRing(R, NonNegativeInteger)

MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

CoercibleFrom(R)

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

RetractableTo(Fraction(Integer))

RetractableTo(R)

AbelianGroup

NonAssociativeAlgebra(R)