# DifferentialPolynomialCategory(R, S, V, E)ΒΆ

dpolcat.spad line 175 [edit on github]

R: Ring

S: OrderedSet

DifferentialPolynomialCategory is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition, it implements defaults for the basic functions. The functions order and weight are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are leader, initial, separant, differentialVariables, and isobaric?. Furthermore, if the ground ring is a differential ring, then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by eval. A convenient way of referencing derivatives is provided by the functions makeVariable. To construct a domain using this constructor, one needs to provide a ground ring `R`

, an ordered set `S`

of differential indeterminates, a ranking `V`

on the set of derivatives of the differential indeterminates, and a set `E`

of exponents in bijection with the set of differential monomials in the given differential indeterminates.

- 0: %
from AbelianMonoid

- 1: %
from MagmaWithUnit

- *: (%, %) -> %
from Magma

- *: (%, 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, E)

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- annihilate?: (%, %) -> Boolean
from Rng

- antiCommutator: (%, %) -> %

- 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 % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has CharacteristicNonZero

- coefficient: (%, E) -> R
from AbelianMonoidRing(R, E)

- coefficient: (%, List V, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, E, V)

- coefficient: (%, V, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, E, V)

- coefficients: % -> List R
from FreeModuleCategory(R, E)

- coerce: % -> % if R has CommutativeRing
from Algebra %

- coerce: % -> OutputForm
from CoercibleTo OutputForm

- coerce: Fraction Integer -> % if R has Algebra Fraction Integer or R has RetractableTo Fraction Integer
- coerce: Integer -> %
from NonAssociativeRing

- coerce: R -> %
from Algebra R

- coerce: S -> %
from CoercibleFrom S

- coerce: V -> %
from CoercibleFrom V

- commutator: (%, %) -> %
from NonAssociativeRng

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

- construct: List Record(k: E, c: R) -> %
from IndexedProductCategory(R, E)

- constructOrdered: List Record(k: E, c: R) -> %
from IndexedProductCategory(R, E)

- content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, E)

- content: (%, V) -> % if R has GcdDomain
from PolynomialCategory(R, E, 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
- D: (%, List Symbol, List NonNegativeInteger) -> % if R has 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
- D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- D: (%, V) -> %
from PartialDifferentialRing V

- D: (%, V, NonNegativeInteger) -> %
from PartialDifferentialRing V

- degree: % -> E
from AbelianMonoidRing(R, E)

- degree: (%, List V) -> List NonNegativeInteger
from MaybeSkewPolynomialCategory(R, E, V)

- degree: (%, S) -> NonNegativeInteger
`degree(p, s)`

returns the maximum degree of the differential polynomial`p`

viewed as a differential polynomial in the differential indeterminate`s`

alone.- degree: (%, V) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, E, V)

- differentialVariables: % -> List S
`differentialVariables(p)`

returns a list of differential indeterminates occurring in a differential polynomial`p`

.

- differentiate: % -> % if R has DifferentialRing
from DifferentialRing

- differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has 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
- differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- differentiate: (%, V) -> %
from PartialDifferentialRing V

- differentiate: (%, V, NonNegativeInteger) -> %
from PartialDifferentialRing V

- discriminant: (%, V) -> % if R has CommutativeRing
from PolynomialCategory(R, E, 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, E)

- factor: % -> Factored % if R has PolynomialFactorizationExplicit

- factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

- factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

- fmecg: (%, E, R, %) -> %
from FiniteAbelianMonoidRing(R, E)

- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has GcdDomain

- ground?: % -> Boolean
from FiniteAbelianMonoidRing(R, E)

- ground: % -> R
from FiniteAbelianMonoidRing(R, E)

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

- initial: % -> %
`initial(p)`

returns the leading coefficient when the differential polynomial`p`

is written as a univariate polynomial in its leader.

- isExpt: % -> Union(Record(var: V, exponent: NonNegativeInteger), failed)
from PolynomialCategory(R, E, V)

- isobaric?: % -> Boolean
`isobaric?(p)`

returns`true`

if every differential monomial appearing in the differential polynomial`p`

has same weight, and returns`false`

otherwise.

- isPlus: % -> Union(List %, failed)
from PolynomialCategory(R, E, V)

- isTimes: % -> Union(List %, failed)
from PolynomialCategory(R, E, V)

- latex: % -> String
from SetCategory

- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has GcdDomain
from LeftOreRing

- leader: % -> V
`leader(p)`

returns the derivative of the highest rank appearing in the differential polynomial`p`

Note: an error occurs if`p`

is in the ground ring.

- leadingCoefficient: % -> R
from IndexedProductCategory(R, E)

- leadingMonomial: % -> %
from IndexedProductCategory(R, E)

- leadingSupport: % -> E
from IndexedProductCategory(R, E)

- leadingTerm: % -> Record(k: E, c: R)
from IndexedProductCategory(R, E)

- leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- leftPower: (%, PositiveInteger) -> %
from Magma

- leftRecip: % -> Union(%, failed)
from MagmaWithUnit

- linearExtend: (E -> R, %) -> R if R has CommutativeRing
from FreeModuleCategory(R, E)

- listOfTerms: % -> List Record(k: E, c: R)
from IndexedDirectProductCategory(R, E)

- mainVariable: % -> Union(V, failed)
from MaybeSkewPolynomialCategory(R, E, V)

- makeVariable: % -> NonNegativeInteger -> % if R has DifferentialRing
`makeVariable(p)`

views`p`

as an element of a differential ring, in such a way that the`n`

-th derivative of`p`

may be simply referenced as`z`

.`n`

where`z`

`:=`

makeVariable(`p`

). Note: In the interpreter,`z`

is given as an internal map, which may be ignored.

- makeVariable: S -> NonNegativeInteger -> %
`makeVariable(s)`

views`s`

as a differential indeterminate, in such a way that the`n`

-th derivative of`s`

may be simply referenced as`z`

.`n`

where`z`

`:=`

makeVariable(`s`

). Note: In the interpreter,`z`

is given as an internal map, which may be ignored.

- map: (R -> R, %) -> %
from IndexedProductCategory(R, E)

- mapExponents: (E -> E, %) -> %
from FiniteAbelianMonoidRing(R, E)

- minimumDegree: % -> E
from FiniteAbelianMonoidRing(R, E)

- minimumDegree: (%, List V) -> List NonNegativeInteger
from PolynomialCategory(R, E, V)

- minimumDegree: (%, V) -> NonNegativeInteger
from PolynomialCategory(R, E, V)

- monicDivide: (%, %, V) -> Record(quotient: %, remainder: %)
from PolynomialCategory(R, E, V)

- monomial?: % -> Boolean
from IndexedProductCategory(R, E)

- monomial: (%, List V, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, E, V)

- monomial: (%, V, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, E, V)

- monomial: (R, E) -> %
from IndexedProductCategory(R, E)

- monomials: % -> List %
from MaybeSkewPolynomialCategory(R, E, V)

- multivariate: (SparseUnivariatePolynomial %, V) -> %
from PolynomialCategory(R, E, V)

- multivariate: (SparseUnivariatePolynomial R, V) -> %
from PolynomialCategory(R, E, V)

- numberOfMonomials: % -> NonNegativeInteger
from IndexedDirectProductCategory(R, E)

- one?: % -> Boolean
from MagmaWithUnit

- opposite?: (%, %) -> Boolean
from AbelianMonoid

- order: % -> NonNegativeInteger
`order(p)`

returns the order of the differential polynomial`p`

, which is the maximum number of differentiations of a differential indeterminate, among all those appearing in`p`

.

- order: (%, S) -> NonNegativeInteger
`order(p, s)`

returns the order of the differential polynomial`p`

in differential indeterminate`s`

.

- 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

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

- pomopo!: (%, R, E, %) -> %
from FiniteAbelianMonoidRing(R, E)

- prime?: % -> Boolean if R has PolynomialFactorizationExplicit

- primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(R, E, V)

- primitivePart: % -> % if R has GcdDomain
from PolynomialCategory(R, E, V)

- primitivePart: (%, V) -> % if R has GcdDomain
from PolynomialCategory(R, E, V)

- recip: % -> Union(%, failed)
from MagmaWithUnit

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

- reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer
- reducedSystem: Matrix % -> Matrix R
from LinearlyExplicitOver R

- reductum: % -> %
from IndexedProductCategory(R, E)

- resultant: (%, %, V) -> % if R has CommutativeRing
from PolynomialCategory(R, E, 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: % -> 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(V, failed)
from RetractableTo V

- rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- rightPower: (%, PositiveInteger) -> %
from Magma

- rightRecip: % -> Union(%, failed)
from MagmaWithUnit

- sample: %
from AbelianMonoid

- separant: % -> %
`separant(p)`

returns the partial derivative of the differential polynomial`p`

with respect to its leader.

- smaller?: (%, %) -> Boolean if R has Comparable
from Comparable

- solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has PolynomialFactorizationExplicit

- squareFree: % -> Factored % if R has GcdDomain
from PolynomialCategory(R, E, V)

- squareFreePart: % -> % if R has GcdDomain
from PolynomialCategory(R, E, V)

- squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit

- subtractIfCan: (%, %) -> Union(%, failed)

- support: % -> List E
from FreeModuleCategory(R, E)

- totalDegree: % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, E, V)

- totalDegree: (%, List V) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, E, V)

- totalDegreeSorted: (%, List V) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, E, 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, E, V)

- univariate: (%, V) -> SparseUnivariatePolynomial %
from PolynomialCategory(R, E, V)

- variables: % -> List V
from MaybeSkewPolynomialCategory(R, E, V)

- weight: % -> NonNegativeInteger
`weight(p)`

returns the maximum weight of all differential monomials appearing in the differential polynomial`p`

.

- weight: (%, S) -> NonNegativeInteger
`weight(p, s)`

returns the maximum weight of all differential monomials appearing in the differential polynomial`p`

when`p`

is viewed as a differential polynomial in the differential indeterminate`s`

alone.

- weights: % -> List NonNegativeInteger
`weights(p)`

returns a list of weights of differential monomials appearing in differential polynomial`p`

.

- weights: (%, S) -> List NonNegativeInteger
`weights(p, s)`

returns a list of weights of differential monomials appearing in the differential polynomial`p`

when`p`

is viewed as a differential polynomial in the differential indeterminate`s`

alone.

- zero?: % -> Boolean
from AbelianMonoid

AbelianMonoidRing(R, E)

Algebra % if R has CommutativeRing

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if R has CommutativeRing

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

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

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

DifferentialRing if R has DifferentialRing

EntireRing if R has EntireRing

Evalable %

FiniteAbelianMonoidRing(R, E)

FreeModuleCategory(R, E)

Hashable if R has Hashable and V has Hashable

IndexedDirectProductCategory(R, E)

IndexedProductCategory(R, E)

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 Fraction Integer if R has Algebra Fraction Integer

LeftOreRing if R has GcdDomain

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

MaybeSkewPolynomialCategory(R, E, V)

Module % if R has CommutativeRing

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

NonAssociativeAlgebra % if R has CommutativeRing

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

NonAssociativeAlgebra R if R has CommutativeRing

noZeroDivisors if R has EntireRing

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol

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, E, V)

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

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

RightModule Integer if R has LinearlyExplicitOver Integer

TwoSidedRecip if R has CommutativeRing

UniqueFactorizationDomain if R has PolynomialFactorizationExplicit