JetBundlePolynomial(R, JB)

jet.spad line 6455 [edit on github]

• R : Ring
• JB : JetBundleCategory

JetBundlePolynomial implements polynomial sections over a jet bundle. The order is not fixed, thus jet variables of any order can appear.

* : (%, %) -> %

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(JB))

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

D : (%, JB) -> %

from PartialDifferentialRing(JB)

D : (%, JB, NonNegativeInteger) -> %

from PartialDifferentialRing(JB)

D : (%, List(JB)) -> %

from PartialDifferentialRing(JB)

D : (%, List(JB), List(NonNegativeInteger)) -> %

from PartialDifferentialRing(JB)

D : (%, List(Symbol)) -> %

from PartialDifferentialRing(Symbol)

D : (%, List(Symbol), List(NonNegativeInteger)) -> %

from PartialDifferentialRing(Symbol)

D : (%, Symbol) -> %

from PartialDifferentialRing(Symbol)

D : (%, Symbol, NonNegativeInteger) -> %

from PartialDifferentialRing(Symbol)

P : List(NonNegativeInteger) -> %

from JetBundleFunctionCategory(JB)

P : NonNegativeInteger -> %

from JetBundleFunctionCategory(JB)

P : (PositiveInteger, List(NonNegativeInteger)) -> %

from JetBundleFunctionCategory(JB)

P : (PositiveInteger, NonNegativeInteger) -> %

from JetBundleFunctionCategory(JB)

U : () -> %

from JetBundleFunctionCategory(JB)

U : PositiveInteger -> %

from JetBundleFunctionCategory(JB)

X : () -> %

from JetBundleFunctionCategory(JB)

X : PositiveInteger -> %

from JetBundleFunctionCategory(JB)

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

autoReduce : List(%) -> List(%)

from JetBundleFunctionCategory(JB)

binomThmExpt : (%, %, NonNegativeInteger) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

charthRoot : % -> Union(%, "failed") if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has CharacteristicNonZero

from PolynomialFactorizationExplicit

class : % -> NonNegativeInteger

from JetBundleFunctionCategory(JB)

coefficient : (%, JB, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

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

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

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

from AbelianMonoidRing(R, IndexedExponents(JB))

coefficients : % -> List(R)

from FreeModuleCategory(R, IndexedExponents(JB))

coerce : % -> %

from Algebra(%)

coerce : JB -> %

from CoercibleFrom(JB)

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

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

from PolynomialFactorizationExplicit

const? : % -> Boolean

from JetBundleFunctionCategory(JB)

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

from IndexedProductCategory(R, IndexedExponents(JB))

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

from IndexedProductCategory(R, IndexedExponents(JB))

content : (%, JB) -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(JB), JB)

content : % -> R if R has GcdDomain

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

convert : % -> InputForm if R has ConvertibleTo(InputForm) and JB has ConvertibleTo(InputForm)

from ConvertibleTo(InputForm)

convert : % -> Pattern(Float) if R has ConvertibleTo(Pattern(Float)) and JB has ConvertibleTo(Pattern(Float))

from ConvertibleTo(Pattern(Float))

convert : % -> Pattern(Integer) if R has ConvertibleTo(Pattern(Integer)) and JB has ConvertibleTo(Pattern(Integer))

from ConvertibleTo(Pattern(Integer))

dSubst : (%, JB, %) -> %

from JetBundleFunctionCategory(JB)

degree : % -> IndexedExponents(JB)

from AbelianMonoidRing(R, IndexedExponents(JB))

degree : (%, List(JB)) -> List(NonNegativeInteger)

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

degree : (%, JB) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

denominator : % -> %

from JetBundleFunctionCategory(JB)

differentiate : (%, JB) -> %

from PartialDifferentialRing(JB)

differentiate : (%, JB, NonNegativeInteger) -> %

from PartialDifferentialRing(JB)

differentiate : (%, List(JB)) -> %

from PartialDifferentialRing(JB)

differentiate : (%, List(JB), List(NonNegativeInteger)) -> %

from PartialDifferentialRing(JB)

differentiate : (%, List(Symbol)) -> %

from PartialDifferentialRing(Symbol)

differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> %

from PartialDifferentialRing(Symbol)

differentiate : (%, Symbol) -> %

from PartialDifferentialRing(Symbol)

differentiate : (%, Symbol, NonNegativeInteger) -> %

from PartialDifferentialRing(Symbol)

dimension : (List(%), SparseEchelonMatrix(JB, %), NonNegativeInteger) -> NonNegativeInteger

from JetBundleFunctionCategory(JB)

discriminant : (%, JB) -> % if R has CommutativeRing

from PolynomialCategory(R, IndexedExponents(JB), JB)

eval : (%, %, %) -> %

from InnerEvalable(%, %)

eval : (%, JB, %) -> %

from InnerEvalable(JB, %)

eval : (%, JB, R) -> %

from InnerEvalable(JB, R)

eval : (%, Equation(%)) -> %

from Evalable(%)

eval : (%, List(%), List(%)) -> %

from InnerEvalable(%, %)

eval : (%, List(JB), List(%)) -> %

from InnerEvalable(JB, %)

eval : (%, List(JB), List(R)) -> %

from InnerEvalable(JB, R)

eval : (%, List(Equation(%))) -> %

from Evalable(%)

exquo : (%, %) -> Union(%, "failed")

from EntireRing

exquo : (%, R) -> Union(%, "failed") if R has EntireRing

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

extractSymbol : SparseEchelonMatrix(JB, %) -> SparseEchelonMatrix(JB, %)

from JetBundleFunctionCategory(JB)

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(JB), R, %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

formalDiff : (%, List(NonNegativeInteger)) -> %

from JetBundleFunctionCategory(JB)

formalDiff : (%, PositiveInteger) -> %

from JetBundleFunctionCategory(JB)

formalDiff : (List(%), PositiveInteger) -> List(%)

from JetBundleFunctionCategory(JB)

formalDiff2 : (%, PositiveInteger, SparseEchelonMatrix(JB, %)) -> Record(DPhi : %, JVars : List(JB))

from JetBundleFunctionCategory(JB)

formalDiff2 : (List(%), PositiveInteger, SparseEchelonMatrix(JB, %)) -> Record(DSys : List(%), JVars : List(List(JB)))

from JetBundleFunctionCategory(JB)

freeOf? : (%, JB) -> Boolean

from JetBundleFunctionCategory(JB)

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)

from PolynomialFactorizationExplicit

getNotation : () -> Symbol

from JetBundleFunctionCategory(JB)

groebner : List(%) -> List(%) if R has GcdDomain

groebner(lp) computes a Groebner basis for the ideal generated by lp wrt a lexicographic ordering.

ground : % -> R

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

ground? : % -> Boolean

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

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

from Hashable

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

from Hashable

isExpt : % -> Union(Record(var : JB, exponent : NonNegativeInteger), "failed")

from PolynomialCategory(R, IndexedExponents(JB), JB)

isPlus : % -> Union(List(%), "failed")

from PolynomialCategory(R, IndexedExponents(JB), JB)

isTimes : % -> Union(List(%), "failed")

from PolynomialCategory(R, IndexedExponents(JB), JB)

jacobiMatrix : List(%) -> SparseEchelonMatrix(JB, %)

from JetBundleFunctionCategory(JB)

jacobiMatrix : (List(%), List(List(JB))) -> SparseEchelonMatrix(JB, %)

from JetBundleFunctionCategory(JB)

jetVariables : % -> List(JB)

from JetBundleFunctionCategory(JB)

latex : % -> String

from SetCategory

lcm : (%, %) -> %

from GcdDomain

lcm : List(%) -> %

from GcdDomain

lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %)

from LeftOreRing

leadingCoefficient : % -> R

from IndexedProductCategory(R, IndexedExponents(JB))

leadingDer : % -> JB

from JetBundleFunctionCategory(JB)

leadingMonomial : % -> %

from IndexedProductCategory(R, IndexedExponents(JB))

leadingSupport : % -> IndexedExponents(JB)

from IndexedProductCategory(R, IndexedExponents(JB))

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

from IndexedProductCategory(R, IndexedExponents(JB))

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

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

from FreeModuleCategory(R, IndexedExponents(JB))

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

from IndexedDirectProductCategory(R, IndexedExponents(JB))

mainVariable : % -> Union(JB, "failed")

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

map : (Mapping(R, R), %) -> %

from IndexedProductCategory(R, IndexedExponents(JB))

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

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

minimumDegree : % -> IndexedExponents(JB)

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

minimumDegree : (%, List(JB)) -> List(NonNegativeInteger)

from PolynomialCategory(R, IndexedExponents(JB), JB)

minimumDegree : (%, JB) -> NonNegativeInteger

from PolynomialCategory(R, IndexedExponents(JB), JB)

monicDivide : (%, %, JB) -> Record(quotient : %, remainder : %)

from PolynomialCategory(R, IndexedExponents(JB), JB)

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

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

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

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

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

from IndexedProductCategory(R, IndexedExponents(JB))

monomial? : % -> Boolean

from IndexedProductCategory(R, IndexedExponents(JB))

monomials : % -> List(%)

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

multivariate : (SparseUnivariatePolynomial(%), JB) -> %

from PolynomialCategory(R, IndexedExponents(JB), JB)

multivariate : (SparseUnivariatePolynomial(R), JB) -> %

from PolynomialCategory(R, IndexedExponents(JB), JB)

numDepVar : () -> PositiveInteger

from JetBundleFunctionCategory(JB)

numIndVar : () -> PositiveInteger

from JetBundleFunctionCategory(JB)

numberOfMonomials : % -> NonNegativeInteger

from IndexedDirectProductCategory(R, IndexedExponents(JB))

numerator : % -> %

from JetBundleFunctionCategory(JB)

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

order : % -> NonNegativeInteger

from JetBundleFunctionCategory(JB)

orderDim : (List(%), SparseEchelonMatrix(JB, %), NonNegativeInteger) -> NonNegativeInteger

from JetBundleFunctionCategory(JB)

patternMatch : (%, Pattern(Float), PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if JB has PatternMatchable(Float) and R has PatternMatchable(Float)

from PatternMatchable(Float)

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

from PatternMatchable(Integer)

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

pomopo! : (%, R, IndexedExponents(JB), %) -> %

from FiniteAbelianMonoidRing(R, IndexedExponents(JB))

prime? : % -> Boolean if R has PolynomialFactorizationExplicit

from UniqueFactorizationDomain

primitiveMonomials : % -> List(%)

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

primitivePart : % -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(JB), JB)

primitivePart : (%, JB) -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(JB), JB)

recip : % -> Union(%, "failed")

from MagmaWithUnit

reduceMod : (List(%), List(%)) -> List(%)

from JetBundleFunctionCategory(JB)

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(JB))

resultant : (%, %, JB) -> % if R has CommutativeRing

from PolynomialCategory(R, IndexedExponents(JB), JB)

retract : % -> JB

from RetractableTo(JB)

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(JB, "failed")

from RetractableTo(JB)

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

setNotation : Symbol -> Void

from JetBundleFunctionCategory(JB)

simpMod : (List(%), List(%)) -> List(%)

from JetBundleFunctionCategory(JB)

simpMod : (List(%), SparseEchelonMatrix(JB, %), List(%)) -> Record(Sys : List(%), JM : SparseEchelonMatrix(JB, %), Depend : Union("failed", List(List(NonNegativeInteger))))

from JetBundleFunctionCategory(JB)

simpOne : % -> %

from JetBundleFunctionCategory(JB)

simplify : (List(%), SparseEchelonMatrix(JB, %)) -> Record(Sys : List(%), JM : SparseEchelonMatrix(JB, %), Depend : Union("failed", List(List(NonNegativeInteger))))

from JetBundleFunctionCategory(JB)

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

from Comparable

solveFor : (%, JB) -> Union(%, "failed")

from JetBundleFunctionCategory(JB)

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

from PolynomialFactorizationExplicit

sortLD : List(%) -> List(%)

from JetBundleFunctionCategory(JB)

squareFree : % -> Factored(%) if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(JB), JB)

squareFreePart : % -> % if R has GcdDomain

from PolynomialCategory(R, IndexedExponents(JB), JB)

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

from PolynomialFactorizationExplicit

subst : (%, JB, %) -> %

from JetBundleFunctionCategory(JB)

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

from CancellationAbelianMonoid

support : % -> List(IndexedExponents(JB))

from FreeModuleCategory(R, IndexedExponents(JB))

symbol : List(%) -> SparseEchelonMatrix(JB, %)

from JetBundleFunctionCategory(JB)

totalDegree : % -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

totalDegree : (%, List(JB)) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

totalDegreeSorted : (%, List(JB)) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %)

from EntireRing

univariate : (%, JB) -> SparseUnivariatePolynomial(%)

from PolynomialCategory(R, IndexedExponents(JB), JB)

univariate : % -> SparseUnivariatePolynomial(R)

from PolynomialCategory(R, IndexedExponents(JB), JB)

variables : % -> List(JB)

from MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

CharacteristicNonZero

Module(Fraction(Integer))

NonAssociativeSemiRing

LeftModule(R)

BiModule(%, %)

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

FullyRetractableTo(R)

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

FreeModuleCategory(R, IndexedExponents(JB))

unitsKnown

FullyLinearlyExplicitOver(R)

PatternMatchable(Float)

IndexedProductCategory(R, IndexedExponents(JB))

AbelianSemiGroup

noZeroDivisors

CoercibleFrom(JB)

PartialDifferentialRing(Symbol)

UniqueFactorizationDomain

InnerEvalable(%, %)

SemiGroup

Magma

GcdDomain

IntegralDomain

ConvertibleTo(InputForm)

LeftModule(%)

NonAssociativeRing

CharacteristicZero

Module(R)

MaybeSkewPolynomialCategory(R, IndexedExponents(JB), JB)

Algebra(%)

BiModule(R, R)

RightModule(Fraction(Integer))

Algebra(R)

JetBundleFunctionCategory(JB)

RightModule(R)

NonAssociativeRng

CommutativeRing

LeftOreRing

CancellationAbelianMonoid

RetractableTo(Integer)

SetCategory

LinearlyExplicitOver(R)

AbelianMonoidRing(R, IndexedExponents(JB))

CommutativeStar

VariablesCommuteWithCoefficients

AbelianMonoid

MagmaWithUnit

Comparable

AbelianGroup

FiniteAbelianMonoidRing(R, IndexedExponents(JB))

RightModule(%)

AbelianProductCategory(R)

Hashable

InnerEvalable(JB, %)

Evalable(%)

Module(%)

LinearlyExplicitOver(Integer)

CoercibleTo(OutputForm)

SemiRng

ConvertibleTo(Pattern(Float))

Monoid

PolynomialFactorizationExplicit

NonAssociativeAlgebra(R)

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(Integer)

IndexedDirectProductCategory(R, IndexedExponents(JB))

ConvertibleTo(Pattern(Integer))

CoercibleFrom(Fraction(Integer))

PartialDifferentialRing(JB)

LeftModule(Fraction(Integer))

NonAssociativeSemiRng

CoercibleFrom(R)

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

InnerEvalable(JB, R)

RetractableTo(Fraction(Integer))

RetractableTo(R)

PatternMatchable(Integer)

RetractableTo(JB)

PolynomialCategory(R, IndexedExponents(JB), JB)