DifferentialSparseMultivariatePolynomial(R, S, V)

dpolcat.spad line 407 [edit on github]

• R: Ring

• S: OrderedSet

• V: DifferentialVariableCategory S

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 LeftModule %

*: (%, 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 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 % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has CharacteristicNonZero

from CharacteristicNonZero

coefficient: (%, IndexedExponents V) -> R

from FreeModuleCategory(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 FreeModuleCategory(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 CoercibleFrom S

coerce: SparseMultivariatePolynomial(R, S) -> %

from CoercibleFrom SparseMultivariatePolynomial(R, S)

coerce: V -> %

from CoercibleFrom V

commutator: (%, %) -> %

from NonAssociativeRng

conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

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

from IndexedProductCategory(R, IndexedExponents V)

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

from IndexedProductCategory(R, IndexedExponents V)

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 V has ConvertibleTo InputForm and R has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if V has ConvertibleTo Pattern Float and R has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if V has ConvertibleTo Pattern Integer and R 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 GcdDomain

ground?: % -> Boolean

from FiniteAbelianMonoidRing(R, IndexedExponents V)

ground: % -> R

from FiniteAbelianMonoidRing(R, IndexedExponents V)

hash: % -> SingleInteger if V has Hashable and R has Hashable

from Hashable

hashUpdate!: (HashState, %) -> HashState if V has Hashable and R has Hashable

from Hashable

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 IndexedProductCategory(R, IndexedExponents V)

leadingMonomial: % -> %

from IndexedProductCategory(R, IndexedExponents V)

leadingSupport: % -> IndexedExponents V

from IndexedProductCategory(R, IndexedExponents V)

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

from IndexedProductCategory(R, IndexedExponents V)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

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

from FreeModuleCategory(R, IndexedExponents V)

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

from IndexedDirectProductCategory(R, IndexedExponents V)

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 IndexedProductCategory(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?: % -> Boolean

from IndexedProductCategory(R, IndexedExponents 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 IndexedProductCategory(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 IndexedDirectProductCategory(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 R has PatternMatchable Float and V has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer and V has PatternMatchable Integer

from PatternMatchable Integer

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

from NonAssociativeAlgebra %

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 IndexedProductCategory(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

support: % -> List IndexedExponents V

from FreeModuleCategory(R, IndexedExponents V)

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)

AbelianProductCategory R

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

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom R

CoercibleFrom S

CoercibleFrom SparseMultivariatePolynomial(R, S)

CoercibleFrom V

CoercibleTo OutputForm

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Comparable

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

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

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

FreeModuleCategory(R, IndexedExponents V)

FullyLinearlyExplicitOver R

FullyRetractableTo R

GcdDomain if R has GcdDomain

Hashable if V has Hashable and R has Hashable

IndexedDirectProductCategory(R, IndexedExponents V)

IndexedProductCategory(R, IndexedExponents V)

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

NonAssociativeAlgebra % if R has CommutativeRing

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol

PartialDifferentialRing V

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

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

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if R has CommutativeRing

UniqueFactorizationDomain if R has PolynomialFactorizationExplicit

unitsKnown

VariablesCommuteWithCoefficients