DifferentialSparseMultivariatePolynomial(R, S, V)ΒΆ

dpolcat.spad line 407

DifferentialSparseMultivariatePolynomial implements an ordinary differential polynomial ring by combining a domain belonging to the category DifferentialVariableCategory with the domain SparseMultivariatePolynomial.

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, IndexedExponents V)
=: (%, %) -> 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 V)
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero or % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
coefficient: (%, IndexedExponents V) -> R
from AbelianMonoidRing(R, IndexedExponents V)
coefficient: (%, List V, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
coefficient: (%, V, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
coefficients: % -> List R
from FiniteAbelianMonoidRing(R, IndexedExponents V)
coerce: % -> % if R has CommutativeRing
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
coerce: S -> %
from RetractableTo S
coerce: SparseMultivariatePolynomial(R, S) -> %
from RetractableTo SparseMultivariatePolynomial(R, S)
coerce: V -> %
from RetractableTo V
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents V)
content: (%, V) -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents V, V)
convert: % -> InputForm if R has ConvertibleTo InputForm and V has ConvertibleTo InputForm
from ConvertibleTo InputForm
convert: % -> Pattern Float if R has ConvertibleTo Pattern Float and V has ConvertibleTo Pattern Float
from ConvertibleTo Pattern Float
convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer and V has ConvertibleTo Pattern Integer
from ConvertibleTo Pattern Integer
D: % -> % if R has DifferentialRing
from DifferentialRing
D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, List V) -> %
from PartialDifferentialRing V
D: (%, List V, List NonNegativeInteger) -> %
from PartialDifferentialRing V
D: (%, NonNegativeInteger) -> % if R has DifferentialRing
from DifferentialRing
D: (%, R -> R) -> %
from DifferentialExtension R
D: (%, R -> R, NonNegativeInteger) -> %
from DifferentialExtension R
D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, V) -> %
from PartialDifferentialRing V
D: (%, V, NonNegativeInteger) -> %
from PartialDifferentialRing V
degree: % -> IndexedExponents V
from AbelianMonoidRing(R, IndexedExponents V)
degree: (%, List V) -> List NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
degree: (%, S) -> NonNegativeInteger
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
degree: (%, V) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
differentialVariables: % -> List S
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
differentiate: % -> % if R has DifferentialRing
from DifferentialRing
differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, List V) -> %
from PartialDifferentialRing V
differentiate: (%, List V, List NonNegativeInteger) -> %
from PartialDifferentialRing V
differentiate: (%, NonNegativeInteger) -> % if R has DifferentialRing
from DifferentialRing
differentiate: (%, R -> R) -> %
from DifferentialExtension R
differentiate: (%, R -> R, NonNegativeInteger) -> %
from DifferentialExtension R
differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, V) -> %
from PartialDifferentialRing V
differentiate: (%, V, NonNegativeInteger) -> %
from PartialDifferentialRing V
discriminant: (%, V) -> % if R has CommutativeRing
from PolynomialCategory(R, IndexedExponents V, V)
eval: (%, %, %) -> %
from InnerEvalable(%, %)
eval: (%, Equation %) -> %
from Evalable %
eval: (%, List %, List %) -> %
from InnerEvalable(%, %)
eval: (%, List Equation %) -> %
from Evalable %
eval: (%, List S, List %) -> % if R has DifferentialRing
from InnerEvalable(S, %)
eval: (%, List S, List R) -> % if R has DifferentialRing
from InnerEvalable(S, R)
eval: (%, List V, List %) -> %
from InnerEvalable(V, %)
eval: (%, List V, List R) -> %
from InnerEvalable(V, R)
eval: (%, S, %) -> % if R has DifferentialRing
from InnerEvalable(S, %)
eval: (%, S, R) -> % if R has DifferentialRing
from InnerEvalable(S, R)
eval: (%, V, %) -> %
from InnerEvalable(V, %)
eval: (%, V, R) -> %
from InnerEvalable(V, R)
exquo: (%, %) -> Union(%, failed) if R has EntireRing
from EntireRing
exquo: (%, R) -> Union(%, failed) if R has EntireRing
from FiniteAbelianMonoidRing(R, IndexedExponents V)
factor: % -> Factored % if R has PolynomialFactorizationExplicit
from UniqueFactorizationDomain
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
fmecg: (%, IndexedExponents V, R, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents V)
gcd: (%, %) -> % if R has GcdDomain
from GcdDomain
gcd: List % -> % if R has GcdDomain
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has GcdDomain
from PolynomialFactorizationExplicit
ground: % -> R
from FiniteAbelianMonoidRing(R, IndexedExponents V)
ground?: % -> Boolean
from FiniteAbelianMonoidRing(R, IndexedExponents V)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
initial: % -> %
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
isExpt: % -> Union(Record(var: V, exponent: NonNegativeInteger), failed)
from PolynomialCategory(R, IndexedExponents V, V)
isobaric?: % -> Boolean
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
isPlus: % -> Union(List %, failed)
from PolynomialCategory(R, IndexedExponents V, V)
isTimes: % -> Union(List %, failed)
from PolynomialCategory(R, IndexedExponents V, V)
latex: % -> String
from SetCategory
lcm: (%, %) -> % if R has GcdDomain
from GcdDomain
lcm: List % -> % if R has GcdDomain
from GcdDomain
lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has GcdDomain
from LeftOreRing
leader: % -> V
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
leadingCoefficient: % -> R
from AbelianMonoidRing(R, IndexedExponents V)
leadingMonomial: % -> %
from AbelianMonoidRing(R, IndexedExponents V)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mainVariable: % -> Union(V, failed)
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
makeVariable: % -> NonNegativeInteger -> % if R has DifferentialRing
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
makeVariable: S -> NonNegativeInteger -> %
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
map: (R -> R, %) -> %
from AbelianMonoidRing(R, IndexedExponents V)
mapExponents: (IndexedExponents V -> IndexedExponents V, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents V)
minimumDegree: % -> IndexedExponents V
from FiniteAbelianMonoidRing(R, IndexedExponents V)
minimumDegree: (%, List V) -> List NonNegativeInteger
from PolynomialCategory(R, IndexedExponents V, V)
minimumDegree: (%, V) -> NonNegativeInteger
from PolynomialCategory(R, IndexedExponents V, V)
monicDivide: (%, %, V) -> Record(quotient: %, remainder: %)
from PolynomialCategory(R, IndexedExponents V, V)
monomial: (%, List V, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
monomial: (%, V, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
monomial: (R, IndexedExponents V) -> %
from AbelianMonoidRing(R, IndexedExponents V)
monomial?: % -> Boolean
from AbelianMonoidRing(R, IndexedExponents V)
monomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
multivariate: (SparseUnivariatePolynomial %, V) -> %
from PolynomialCategory(R, IndexedExponents V, V)
multivariate: (SparseUnivariatePolynomial R, V) -> %
from PolynomialCategory(R, IndexedExponents V, V)
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, IndexedExponents V)
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
order: % -> NonNegativeInteger
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
order: (%, S) -> NonNegativeInteger
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if V has PatternMatchable Float and R has PatternMatchable Float
from PatternMatchable Float
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if V has PatternMatchable Integer and R has PatternMatchable Integer
from PatternMatchable Integer
pomopo!: (%, R, IndexedExponents V, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents V)
prime?: % -> Boolean if R has PolynomialFactorizationExplicit
from UniqueFactorizationDomain
primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
primitivePart: % -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents V, V)
primitivePart: (%, V) -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents V, V)
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, IndexedExponents V)
resultant: (%, %, V) -> % if R has CommutativeRing
from PolynomialCategory(R, IndexedExponents V, V)
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
retract: % -> S
from RetractableTo S
retract: % -> SparseMultivariatePolynomial(R, S)
from RetractableTo SparseMultivariatePolynomial(R, S)
retract: % -> V
from RetractableTo V
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
retractIfCan: % -> Union(S, failed)
from RetractableTo S
retractIfCan: % -> Union(SparseMultivariatePolynomial(R, S), failed)
from RetractableTo SparseMultivariatePolynomial(R, S)
retractIfCan: % -> Union(V, failed)
from RetractableTo V
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
separant: % -> %
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
smaller?: (%, %) -> Boolean if R has Comparable
from Comparable
solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
squareFree: % -> Factored % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents V, V)
squareFreePart: % -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents V, V)
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
totalDegree: % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
totalDegree: (%, List V) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
totalDegreeSorted: (%, List V) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
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
univariate: % -> SparseUnivariatePolynomial R
from PolynomialCategory(R, IndexedExponents V, V)
univariate: (%, V) -> SparseUnivariatePolynomial %
from PolynomialCategory(R, IndexedExponents V, V)
variables: % -> List V
from MaybeSkewPolynomialCategory(R, IndexedExponents V, V)
weight: % -> NonNegativeInteger
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
weight: (%, S) -> NonNegativeInteger
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
weights: % -> List NonNegativeInteger
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
weights: (%, S) -> List NonNegativeInteger
from DifferentialPolynomialCategory(R, S, V, IndexedExponents V)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, IndexedExponents V)

AbelianSemiGroup

Algebra % if R has CommutativeRing

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if 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

CoercibleTo OutputForm

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Comparable

ConvertibleTo InputForm if R has ConvertibleTo InputForm and V has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float and V has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer and V has ConvertibleTo Pattern Integer

DifferentialExtension R

DifferentialPolynomialCategory(R, S, V, IndexedExponents V)

DifferentialRing if R has DifferentialRing

EntireRing if R has EntireRing

Evalable %

FiniteAbelianMonoidRing(R, IndexedExponents V)

FullyLinearlyExplicitOver R

FullyRetractableTo R

GcdDomain if R has GcdDomain

InnerEvalable(%, %)

InnerEvalable(S, %) if R has DifferentialRing

InnerEvalable(S, R) if R has DifferentialRing

InnerEvalable(V, %)

InnerEvalable(V, R)

IntegralDomain if R has IntegralDomain

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

LeftOreRing if R has GcdDomain

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

MaybeSkewPolynomialCategory(R, IndexedExponents V, V)

Module % if R has CommutativeRing

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol

PartialDifferentialRing V

PatternMatchable Float if V has PatternMatchable Float and R has PatternMatchable Float

PatternMatchable Integer if V has PatternMatchable Integer and R has PatternMatchable Integer

PolynomialCategory(R, IndexedExponents V, V)

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RetractableTo S

RetractableTo SparseMultivariatePolynomial(R, S)

RetractableTo V

RightModule %

RightModule Fraction Integer if R has Algebra Fraction Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

UniqueFactorizationDomain if R has PolynomialFactorizationExplicit

unitsKnown

VariablesCommuteWithCoefficients