JetBundlePolynomial(R, JB)ΒΆ

jet.spad line 6459

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

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 JB)
=: (%, %) -> Boolean
from BasicType
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
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: (%, IndexedExponents JB) -> R
from AbelianMonoidRing(R, IndexedExponents JB)
coefficient: (%, JB, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents JB, JB)
coefficient: (%, List JB, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents JB, JB)
coefficients: % -> List R
from FiniteAbelianMonoidRing(R, IndexedExponents JB)
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has Algebra Fraction Integer or R has RetractableTo Fraction Integer
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: JB -> %
from RetractableTo JB
coerce: R -> %
from Algebra R
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
const?: % -> Boolean
from JetBundleFunctionCategory JB
content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents JB)
content: (%, JB) -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents JB, JB)
convert: % -> InputForm if JB has ConvertibleTo InputForm and R has ConvertibleTo InputForm
from ConvertibleTo InputForm
convert: % -> Pattern Float if JB has ConvertibleTo Pattern Float and R has ConvertibleTo Pattern Float
from ConvertibleTo Pattern Float
convert: % -> Pattern Integer if JB has ConvertibleTo Pattern Integer and R has ConvertibleTo Pattern Integer
from ConvertibleTo Pattern Integer
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
degree: % -> IndexedExponents JB
from AbelianMonoidRing(R, IndexedExponents JB)
degree: (%, JB) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents JB, JB)
degree: (%, List JB) -> List 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)
dSubst: (%, JB, %) -> %
from JetBundleFunctionCategory JB
eval: (%, %, %) -> %
from InnerEvalable(%, %)
eval: (%, Equation %) -> %
from Evalable %
eval: (%, JB, %) -> %
from InnerEvalable(JB, %)
eval: (%, JB, R) -> %
from InnerEvalable(JB, R)
eval: (%, List %, List %) -> %
from InnerEvalable(%, %)
eval: (%, List Equation %) -> %
from Evalable %
eval: (%, List JB, List %) -> %
from InnerEvalable(JB, %)
eval: (%, List JB, List R) -> %
from InnerEvalable(JB, R)
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)
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
formalDiff: (%, List NonNegativeInteger) -> %
from JetBundleFunctionCategory JB
formalDiff: (%, PositiveInteger) -> %
from JetBundleFunctionCategory JB
formalDiff: (List %, PositiveInteger) -> List %
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
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
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 %, List List JB) -> SparseEchelonMatrix(JB, %)
from JetBundleFunctionCategory JB
jacobiMatrix: List % -> 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 AbelianMonoidRing(R, IndexedExponents JB)
leadingDer: % -> JB
from JetBundleFunctionCategory JB
leadingMonomial: % -> %
from AbelianMonoidRing(R, IndexedExponents JB)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mainVariable: % -> Union(JB, failed)
from MaybeSkewPolynomialCategory(R, IndexedExponents JB, JB)
map: (R -> R, %) -> %
from AbelianMonoidRing(R, IndexedExponents JB)
mapExponents: (IndexedExponents JB -> IndexedExponents JB, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents JB)
minimumDegree: % -> IndexedExponents JB
from FiniteAbelianMonoidRing(R, IndexedExponents JB)
minimumDegree: (%, JB) -> NonNegativeInteger
from PolynomialCategory(R, IndexedExponents JB, JB)
minimumDegree: (%, List JB) -> List 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 AbelianMonoidRing(R, IndexedExponents JB)
monomial?: % -> Boolean
from AbelianMonoidRing(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)
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, IndexedExponents JB)
numDepVar: () -> PositiveInteger
from JetBundleFunctionCategory JB
numerator: % -> %
from JetBundleFunctionCategory JB
numIndVar: () -> PositiveInteger
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
P: (PositiveInteger, List NonNegativeInteger) -> %
from JetBundleFunctionCategory JB
P: (PositiveInteger, NonNegativeInteger) -> %
from JetBundleFunctionCategory JB
P: List NonNegativeInteger -> %
from JetBundleFunctionCategory JB
P: 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
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
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
reduceMod: (List %, List %) -> List %
from JetBundleFunctionCategory JB
reductum: % -> %
from AbelianMonoidRing(R, IndexedExponents JB)
resultant: (%, %, JB) -> % if R has CommutativeRing
from PolynomialCategory(R, IndexedExponents JB, JB)
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer
retract: % -> JB
from RetractableTo JB
retract: % -> R
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
retractIfCan: % -> Union(JB, failed)
from RetractableTo JB
retractIfCan: % -> Union(R, failed)
from RetractableTo R
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
setNotation: Symbol -> Void
from JetBundleFunctionCategory JB
simplify: (List %, SparseEchelonMatrix(JB, %)) -> Record(Sys: List %, JM: SparseEchelonMatrix(JB, %), Depend: Union(failed, List List NonNegativeInteger))
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
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
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)
U: () -> %
from JetBundleFunctionCategory JB
U: PositiveInteger -> %
from JetBundleFunctionCategory JB
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
univariate: % -> SparseUnivariatePolynomial R
from PolynomialCategory(R, IndexedExponents JB, JB)
univariate: (%, JB) -> SparseUnivariatePolynomial %
from PolynomialCategory(R, IndexedExponents JB, JB)
variables: % -> List JB
from MaybeSkewPolynomialCategory(R, IndexedExponents JB, JB)
X: () -> %
from JetBundleFunctionCategory JB
X: PositiveInteger -> %
from JetBundleFunctionCategory JB
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, IndexedExponents JB)

AbelianSemiGroup

Algebra %

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

CommutativeStar

Comparable if R has Comparable

ConvertibleTo InputForm if JB has ConvertibleTo InputForm and R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if JB has ConvertibleTo Pattern Float and R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if JB has ConvertibleTo Pattern Integer and R has ConvertibleTo Pattern Integer

EntireRing

Evalable %

FiniteAbelianMonoidRing(R, IndexedExponents JB)

FullyLinearlyExplicitOver R

FullyRetractableTo R

GcdDomain

InnerEvalable(%, %)

InnerEvalable(JB, %)

InnerEvalable(JB, R)

IntegralDomain

JetBundleFunctionCategory JB

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

LeftOreRing

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

MaybeSkewPolynomialCategory(R, IndexedExponents JB, JB)

Module %

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PartialDifferentialRing JB

PartialDifferentialRing Symbol

PatternMatchable Float if JB has PatternMatchable Float and R has PatternMatchable Float

PatternMatchable Integer if JB has PatternMatchable Integer and R has PatternMatchable Integer

PolynomialCategory(R, IndexedExponents JB, JB)

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo JB

RetractableTo R

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