# Polynomial RΒΆ

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 LeftModule %

*: (%, Fraction Integer) -> % if R has Algebra Fraction Integer
*: (%, Integer) -> % if R has LinearlyExplicitOver Integer
*: (%, R) -> %

from RightModule R

*: (Fraction Integer, %) -> % if R has Algebra 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
=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associates?: (%, %) -> Boolean if R has EntireRing

from EntireRing

associator: (%, %, %) -> %
binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero or % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
coefficient: (%, IndexedExponents Symbol) -> R
coefficient: (%, List Symbol, List NonNegativeInteger) -> %
coefficient: (%, Symbol, NonNegativeInteger) -> %
coefficients: % -> List R
coerce: % -> % if R has CommutativeRing

from Algebra %

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

from Algebra R

coerce: Symbol -> %
commutator: (%, %) -> %
conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
construct: List Record(k: IndexedExponents Symbol, c: R) -> %
constructOrdered: List Record(k: IndexedExponents Symbol, c: R) -> %
content: % -> R if R has GcdDomain
content: (%, Symbol) -> % if R has GcdDomain
convert: % -> InputForm if R has ConvertibleTo InputForm
convert: % -> Pattern Float if R has ConvertibleTo Pattern Float
convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer
D: (%, List Symbol) -> %
D: (%, List Symbol, List NonNegativeInteger) -> %
D: (%, Symbol) -> %
D: (%, Symbol, NonNegativeInteger) -> %
degree: % -> IndexedExponents Symbol
degree: (%, List Symbol) -> List NonNegativeInteger
degree: (%, Symbol) -> NonNegativeInteger
differentiate: (%, List Symbol) -> %
differentiate: (%, List Symbol, List NonNegativeInteger) -> %
differentiate: (%, Symbol) -> %
differentiate: (%, Symbol, NonNegativeInteger) -> %
discriminant: (%, Symbol) -> % if R has CommutativeRing
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
factor: % -> Factored % if R has PolynomialFactorizationExplicit
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
fmecg: (%, IndexedExponents Symbol, R, %) -> %
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
ground: % -> R
hash: % -> SingleInteger if R has Hashable

from Hashable

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

from Hashable

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

leadingTerm: % -> Record(k: IndexedExponents Symbol, c: R)
leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

linearExtend: (IndexedExponents Symbol -> R, %) -> R if R has CommutativeRing
listOfTerms: % -> List Record(k: IndexedExponents Symbol, c: R)
mainVariable: % -> Union(Symbol, failed)
map: (R -> R, %) -> %
mapExponents: (IndexedExponents Symbol -> IndexedExponents Symbol, %) -> %
minimumDegree: % -> IndexedExponents Symbol
minimumDegree: (%, List Symbol) -> List NonNegativeInteger
minimumDegree: (%, Symbol) -> NonNegativeInteger
monicDivide: (%, %, Symbol) -> Record(quotient: %, remainder: %)
monomial?: % -> Boolean
monomial: (%, List Symbol, List NonNegativeInteger) -> %
monomial: (%, Symbol, NonNegativeInteger) -> %
monomial: (R, IndexedExponents Symbol) -> %
monomials: % -> List %
multivariate: (SparseUnivariatePolynomial %, Symbol) -> %
multivariate: (SparseUnivariatePolynomial R, Symbol) -> %
numberOfMonomials: % -> NonNegativeInteger
one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer
plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing or R has Algebra Fraction Integer
pomopo!: (%, R, IndexedExponents Symbol, %) -> %
prime?: % -> Boolean if R has PolynomialFactorizationExplicit
primitiveMonomials: % -> List %
primitivePart: % -> % if R has GcdDomain
primitivePart: (%, Symbol) -> % if R has GcdDomain
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: % -> %
resultant: (%, %, Symbol) -> % if R has CommutativeRing
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
retract: % -> R

from RetractableTo R

retract: % -> Symbol
retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
retractIfCan: % -> Union(R, failed)

from RetractableTo R

retractIfCan: % -> Union(Symbol, failed)
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
squareFree: % -> Factored % if R has GcdDomain
squareFreePart: % -> % if R has GcdDomain
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
subtractIfCan: (%, %) -> Union(%, failed)
support: % -> List IndexedExponents Symbol
totalDegree: % -> NonNegativeInteger
totalDegree: (%, List Symbol) -> NonNegativeInteger
totalDegreeSorted: (%, List Symbol) -> NonNegativeInteger
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
univariate: (%, Symbol) -> SparseUnivariatePolynomial %
variables: % -> List Symbol
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra % if R has CommutativeRing

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Comparable

EntireRing if R has EntireRing

GcdDomain if R has GcdDomain

Hashable if R has Hashable

InnerEvalable(%, %)

InnerEvalable(Symbol, %)

InnerEvalable(Symbol, R)

IntegralDomain if R has IntegralDomain

LeftOreRing if R has GcdDomain

Magma

MagmaWithUnit

Module % if R has CommutativeRing

Module R if R has CommutativeRing

Monoid

NonAssociativeAlgebra % if R has CommutativeRing

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if R has CommutativeRing

unitsKnown

VariablesCommuteWithCoefficients