Polynomial RΒΆ

multpoly.spad line 1

This type is the basic representation of sparse recursive multivariate polynomials whose variables are arbitrary symbols. The ordering is alphabetic determined by the Symbol type. The coefficient ring may be non commutative, but the variables are assumed to commute.

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 Symbol)
=: (%, %) -> 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 Symbol)
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
from CharacteristicNonZero
coefficient: (%, IndexedExponents Symbol) -> R
from AbelianMonoidRing(R, IndexedExponents Symbol)
coefficient: (%, List Symbol, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
coefficient: (%, Symbol, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
coefficients: % -> List R
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
coerce: % -> % if R has IntegralDomain
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: R -> %
from Algebra R
coerce: Symbol -> %
from RetractableTo Symbol
commutator: (%, %) -> %
from NonAssociativeRng
content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
content: (%, Symbol) -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
convert: % -> InputForm if R has ConvertibleTo InputForm
from ConvertibleTo InputForm
convert: % -> Pattern Float if R has ConvertibleTo Pattern Float
from ConvertibleTo Pattern Float
convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer
from ConvertibleTo Pattern Integer
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 Symbol
from AbelianMonoidRing(R, IndexedExponents Symbol)
degree: (%, List Symbol) -> List NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
degree: (%, Symbol) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
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
discriminant: (%, Symbol) -> % if R has CommutativeRing
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
eval: (%, %, %) -> %
from InnerEvalable(%, %)
eval: (%, Equation %) -> %
from Evalable %
eval: (%, List %, List %) -> %
from InnerEvalable(%, %)
eval: (%, List Equation %) -> %
from Evalable %
eval: (%, List Symbol, List %) -> %
from InnerEvalable(Symbol, %)
eval: (%, List Symbol, List R) -> %
from InnerEvalable(Symbol, R)
eval: (%, Symbol, %) -> %
from InnerEvalable(Symbol, %)
eval: (%, Symbol, R) -> %
from InnerEvalable(Symbol, R)
exquo: (%, %) -> Union(%, failed) if R has EntireRing
from EntireRing
exquo: (%, R) -> Union(%, failed) if R has EntireRing
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
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 Symbol, R, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
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 Symbol)
ground?: % -> Boolean
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
integrate: (%, Symbol) -> % if R has Algebra Fraction Integer
integrate(p, x) computes the integral of p*dx, i.e. integrates the polynomial p with respect to the variable x.
isExpt: % -> Union(Record(var: Symbol, exponent: NonNegativeInteger), failed)
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
isPlus: % -> Union(List %, failed)
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
isTimes: % -> Union(List %, failed)
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
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
leadingCoefficient: % -> R
from AbelianMonoidRing(R, IndexedExponents Symbol)
leadingMonomial: % -> %
from AbelianMonoidRing(R, IndexedExponents Symbol)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mainVariable: % -> Union(Symbol, failed)
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
map: (R -> R, %) -> %
from AbelianMonoidRing(R, IndexedExponents Symbol)
mapExponents: (IndexedExponents Symbol -> IndexedExponents Symbol, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
minimumDegree: % -> IndexedExponents Symbol
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
minimumDegree: (%, List Symbol) -> List NonNegativeInteger
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
minimumDegree: (%, Symbol) -> NonNegativeInteger
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
monicDivide: (%, %, Symbol) -> Record(quotient: %, remainder: %)
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
monomial: (%, List Symbol, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
monomial: (%, Symbol, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
monomial: (R, IndexedExponents Symbol) -> %
from AbelianMonoidRing(R, IndexedExponents Symbol)
monomial?: % -> Boolean
from AbelianMonoidRing(R, IndexedExponents Symbol)
monomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
multivariate: (SparseUnivariatePolynomial %, Symbol) -> %
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
multivariate: (SparseUnivariatePolynomial R, Symbol) -> %
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float
from PatternMatchable Float
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer
from PatternMatchable Integer
pomopo!: (%, R, IndexedExponents Symbol, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents Symbol)
prime?: % -> Boolean if R has PolynomialFactorizationExplicit
from UniqueFactorizationDomain
primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
primitivePart: % -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
primitivePart: (%, Symbol) -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
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 Symbol)
resultant: (%, %, Symbol) -> % if R has CommutativeRing
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
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: % -> Symbol
from RetractableTo Symbol
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(Symbol, failed)
from RetractableTo Symbol
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
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 Symbol, Symbol)
squareFreePart: % -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
totalDegree: % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
totalDegree: (%, List Symbol) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
totalDegreeSorted: (%, List Symbol) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
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 Symbol, Symbol)
univariate: (%, Symbol) -> SparseUnivariatePolynomial %
from PolynomialCategory(R, IndexedExponents Symbol, Symbol)
variables: % -> List Symbol
from MaybeSkewPolynomialCategory(R, IndexedExponents Symbol, Symbol)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, IndexedExponents Symbol)

AbelianSemiGroup

Algebra % if R has IntegralDomain

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

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

EntireRing if R has EntireRing

Evalable %

FiniteAbelianMonoidRing(R, IndexedExponents Symbol)

FullyLinearlyExplicitOver R

FullyRetractableTo R

GcdDomain if R has GcdDomain

InnerEvalable(%, %)

InnerEvalable(Symbol, %)

InnerEvalable(Symbol, 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 Symbol, Symbol)

Module % if R has IntegralDomain

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

PatternMatchable Float if R has PatternMatchable Float

PatternMatchable Integer if R has PatternMatchable Integer

PolynomialCategory(R, IndexedExponents Symbol, Symbol)

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RetractableTo Symbol

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