NMUnivariatePolynomial(R, x)

jnpoly.spad line 18 [edit on github]

• R: NMRing

• x: Symbol

Univariate polynomial domain using the JL pakage NM Author: G. Vanuxem Date created: mar, 2024

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

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

p * q is the product of p with a rational number q (Fraction(Integer))

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

from RightModule Integer

*: (%, NonNegativeInteger) -> %

p * n is the product of p with a non negative integer.

*: (%, PositiveInteger) -> %

p * n is the product of p with a positive integer.

*: (%, R) -> %

from RightModule R

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

q * p is the product of p with a rational number q (Fraction(Integer))

*: (Integer, %) -> %

from AbelianGroup

*: (NMInteger, %) -> %

from JLObjectRing

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

+: (%, Fraction Integer) -> % if R has QuotientFieldCategory NMInteger

p * q is the sum of p and a rational number q (Fraction(Intger))

+: (Fraction Integer, %) -> % if R has QuotientFieldCategory NMInteger

q * p is the sum of p and a rational number q (Fraction(Intger))

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, R) -> % if R has Field

from AbelianMonoidRing(R, NonNegativeInteger)

=: (%, %) -> 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) -> %

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

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

from PolynomialFactorizationExplicit

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

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

coefficient: (%, NonNegativeInteger) -> R

from FreeModuleCategory(R, NonNegativeInteger)

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

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

coefficients: % -> List R

from FreeModuleCategory(R, NonNegativeInteger)

coerce: % -> %

from Algebra %

coerce: % -> JLObject

from JLObjectType

coerce: % -> OutputForm

from CoercibleTo OutputForm

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

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: R -> %

from CoercibleFrom R

coerce: SingletonAsOrderedSet -> %

from CoercibleFrom SingletonAsOrderedSet

coerce: Variable x -> %

coerce(x) converts the variable x to a NM univariate polynomial.

commutator: (%, %) -> %

from NonAssociativeRng

composite: (%, %) -> Union(%, failed) if R has IntegralDomain

from UnivariatePolynomialCategory R

composite: (Fraction %, %) -> Union(Fraction %, failed) if R has IntegralDomain

from UnivariatePolynomialCategory R

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

from PolynomialFactorizationExplicit

constant?: % -> Boolean

constant?(p) checks whether or not p is a constant polynomial.

constantCoefficient: % -> R

constantCoefficient(p) returns the constant of p.

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

from IndexedProductCategory(R, NonNegativeInteger)

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

from IndexedProductCategory(R, NonNegativeInteger)

content: % -> R if R has GcdDomain

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

convert: % -> InputForm if R has ConvertibleTo InputForm and SingletonAsOrderedSet has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if R has ConvertibleTo Pattern Float and SingletonAsOrderedSet has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer and SingletonAsOrderedSet has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

convert: % -> String

from ConvertibleTo String

cosMinimalPolynomial: (NonNegativeInteger, %) -> % if R has IntegerNumberSystem

cosMinimalPolynomial(n,p) returns the minimal polynomial of 2*cos(2*π/n).

cyclotomicPolynomial: (NonNegativeInteger, %) -> % if R has IntegerNumberSystem

cyclotomicPolynomial(n,p) returns the n-th cyclotomic polynomial Φn.

D: % -> %

from DifferentialRing

D: (%, List SingletonAsOrderedSet) -> %

from PartialDifferentialRing SingletonAsOrderedSet

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

from PartialDifferentialRing SingletonAsOrderedSet

D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> %

from DifferentialRing

D: (%, R -> R) -> %

from DifferentialExtension R

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

from DifferentialExtension R

D: (%, SingletonAsOrderedSet) -> %

from PartialDifferentialRing SingletonAsOrderedSet

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

from PartialDifferentialRing SingletonAsOrderedSet

D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

degree: % -> NonNegativeInteger

from AbelianMonoidRing(R, NonNegativeInteger)

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

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

degree: (%, SingletonAsOrderedSet) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

differentiate: % -> %

from DifferentialRing

differentiate: (%, List SingletonAsOrderedSet) -> %

from PartialDifferentialRing SingletonAsOrderedSet

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

from PartialDifferentialRing SingletonAsOrderedSet

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> %

from DifferentialRing

differentiate: (%, R -> R) -> %

from DifferentialExtension R

differentiate: (%, R -> R, %) -> %

from UnivariatePolynomialCategory R

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

from DifferentialExtension R

differentiate: (%, SingletonAsOrderedSet) -> %

from PartialDifferentialRing SingletonAsOrderedSet

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

from PartialDifferentialRing SingletonAsOrderedSet

differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

discriminant: % -> R if R has CommutativeRing

from UnivariatePolynomialCategory R

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

divide: (%, %) -> Record(quotient: %, remainder: %)

divide(p,q) returns a record (quotient, remainder) of the the Euclidean division of p by q.

divideExponents: (%, NonNegativeInteger) -> Union(%, failed)

from UnivariatePolynomialCategory R

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, Fraction %) -> Fraction % if R has IntegralDomain

from Eltable(Fraction %, Fraction %)

elt: (%, R) -> R

from Eltable(R, R)

elt: (Fraction %, Fraction %) -> Fraction % if R has IntegralDomain

from UnivariatePolynomialCategory R

elt: (Fraction %, R) -> R if R has Field

from UnivariatePolynomialCategory R

etaQExp: (Integer, PositiveInteger, %) -> % if R has IntegerNumberSystem

etaQExp(r,n,p) returns the q-expansion to length n of the Dedekind eta function (without the leading factor q^(1/ 24)) raised to the power r, (q^(−1/24)*η(q))^r.

euclideanSize: % -> NonNegativeInteger if R has Field

from EuclideanDomain

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

from InnerEvalable(%, %)

eval: (%, Equation %) -> %

from Evalable %

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

from InnerEvalable(%, %)

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

from Evalable %

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

from InnerEvalable(SingletonAsOrderedSet, %)

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

from InnerEvalable(SingletonAsOrderedSet, R)

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

from InnerEvalable(SingletonAsOrderedSet, %)

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

from InnerEvalable(SingletonAsOrderedSet, R)

exactDivide: (%, %) -> %

exactDivide(p,q) divides p by q if the division is exact.

expressIdealMember: (List %, %) -> Union(List %, failed) if R has Field

from PrincipalIdealDomain

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

from EntireRing

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

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has Field

from EuclideanDomain

factor: % -> Factored % if R has PolynomialFactorizationExplicit

from UniqueFactorizationDomain

factor: % -> NMFactored %

factor(p) returns the factorization of p using the NMFactored domain.

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

from PolynomialFactorizationExplicit

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

from PolynomialFactorizationExplicit

fmecg: (%, NonNegativeInteger, R, %) -> %

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

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

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

ground: % -> R

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

hash: % -> SingleInteger if R has Hashable

from Hashable

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

from Hashable

init: % if R has StepThrough

from StepThrough

integrate: % -> % if R has Algebra Fraction Integer

from UnivariatePolynomialCategory R

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

jlAbout: % -> Void

from JLObjectType

jlApply: (String, %) -> %

from JLObjectType

jlApply: (String, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %, %) -> %

from JLObjectType

jlApply: (String, %, %, %, %, %) -> %

from JLObjectType

jlDisplay: % -> Void

from JLObjectType

jlId: % -> JLInt64

from JLObjectType

jlNMRing: () -> String

from NMRing

jlObject: () -> String

from NMRing

jlRef: % -> SExpression

from JLObjectType

jlref: String -> %

from JLObjectType

jlType: % -> String

from JLObjectType

jnup: UnivariatePolynomial(x, R) -> %

jnup(p) converts the univariate polynomial p to a NM univariate polynomial.

karatsubaDivide: (%, NonNegativeInteger) -> Record(quotient: %, remainder: %)

from UnivariatePolynomialCategory R

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 IndexedProductCategory(R, NonNegativeInteger)

leadingMonomial: % -> %

from IndexedProductCategory(R, NonNegativeInteger)

leadingSupport: % -> NonNegativeInteger

from IndexedProductCategory(R, NonNegativeInteger)

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

from IndexedProductCategory(R, NonNegativeInteger)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

length: % -> NonNegativeInteger

length(p) returns the number of coefficients in its dense representation. It includes zero coefficients.

linearExtend: (NonNegativeInteger -> R, %) -> R if R has CommutativeRing

from FreeModuleCategory(R, NonNegativeInteger)

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

from IndexedDirectProductCategory(R, NonNegativeInteger)

mainVariable: % -> Union(SingletonAsOrderedSet, failed)

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

makeSUP: % -> SparseUnivariatePolynomial R

from UnivariatePolynomialCategory R

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

from IndexedProductCategory(R, NonNegativeInteger)

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

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

minimalPolynomial: NMAlgebraicNumber -> % if R has QuotientFieldCategory NMInteger or R has IntegerNumberSystem

minimalPolynomial(algn) returns the minimal polynomial of algn. Convenience function.

minimumDegree: % -> NonNegativeInteger

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

minimumDegree: (%, SingletonAsOrderedSet) -> NonNegativeInteger

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

monic?: % -> Boolean

monic?(p) checks whether or not p monic.

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

from UnivariatePolynomialCategory R

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

monomial?: % -> Boolean

monomial?(p) checks whether or not p is a monomial.

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

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

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

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

monomial: (R, NonNegativeInteger) -> %

from IndexedProductCategory(R, NonNegativeInteger)

monomialRecursive?: % -> Boolean

monomialRecursive?(p) checks whether or not p is monomial recurisvely (all scalar types).

monomials: % -> List %

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

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

mullow(p1,p2,n) is the truncated multiplication of p1 and p2 by n.

multiEuclidean: (List %, %) -> Union(List %, failed) if R has Field

from EuclideanDomain

multiplyExponents: (%, NonNegativeInteger) -> %

from UnivariatePolynomialCategory R

multivariate: (SparseUnivariatePolynomial %, SingletonAsOrderedSet) -> %

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

multivariate: (SparseUnivariatePolynomial R, SingletonAsOrderedSet) -> %

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

mutable?: % -> Boolean

from JLObjectType

nextItem: % -> Union(%, failed) if R has StepThrough

from StepThrough

nothing?: % -> Boolean

from JLObjectType

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(R, NonNegativeInteger)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: (%, %) -> NonNegativeInteger if R has IntegralDomain

from UnivariatePolynomialCategory R

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

from PatternMatchable Float

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

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

pomopo!: (%, R, NonNegativeInteger, %) -> %

from FiniteAbelianMonoidRing(R, NonNegativeInteger)

prime?: % -> Boolean if R has PolynomialFactorizationExplicit

from UniqueFactorizationDomain

primitiveMonomials: % -> List %

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

primitivePart: % -> % if R has GcdDomain

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

principalIdeal: List % -> Record(coef: List %, generator: %) if R has Field

from PrincipalIdealDomain

pseudoDivide: (%, %) -> Record(coef: R, quotient: %, remainder: %) if R has IntegralDomain

from UnivariatePolynomialCategory R

pseudoDivide: (%, %) -> Record(quotient: %, remainder: %)

pseudoDivide(p1,p2) returns pseudo-quotient and pseudo-remainder of the pseudo-division of p1 by p2.

pseudoQuotient: (%, %) -> % if R has IntegralDomain

from UnivariatePolynomialCategory R

pseudoRemainder: (%, %) -> %

from UnivariatePolynomialCategory R

quo: (%, %) -> %

a quo b returns the quotient of a and b, forgetting the remainder.

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 IndexedProductCategory(R, NonNegativeInteger)

rem: (%, %) -> %

a rem b returns the remainder of a and b.

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

from UnivariatePolynomialCategory R

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

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

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: % -> SingletonAsOrderedSet

from RetractableTo SingletonAsOrderedSet

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(SingletonAsOrderedSet, failed)

from RetractableTo SingletonAsOrderedSet

reverse: % -> %

reverse the coefficients of p (the leading becomes the trailing) and normalise the resulting polynomial.

reverse: (%, NonNegativeInteger) -> %

reverse the coefficients of p (the leading becomes the trailing) and normalise the resulting polynomial. Adjust the length to n so the resulting polynomial is trucated or padded with zeroes before the leading term if necessary

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

roots: % -> JLVector NMAlgebraicNumber if R has QuotientFieldCategory NMInteger or R has IntegerNumberSystem

roots(p) returns the roots of p. Convenience function.

sample: %

from AbelianMonoid

separate: (%, %) -> Record(primePart: %, commonPart: %) if R has GcdDomain

from UnivariatePolynomialCategory R

shiftLeft: (%, NonNegativeInteger) -> %

from UnivariatePolynomialCategory R

shiftRight: (%, NonNegativeInteger) -> %

from UnivariatePolynomialCategory R

sizeLess?: (%, %) -> Boolean if R has Field

from EuclideanDomain

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

from Comparable

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

from PolynomialFactorizationExplicit

square?: % -> Boolean

square?(p) checks whether or not p is a perfect square.

squareFree: % -> Factored % if R has GcdDomain

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

squareFreePart: % -> % if R has GcdDomain

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

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

from PolynomialFactorizationExplicit

string: % -> String

from JLType

subResultantGcd: (%, %) -> % if R has IntegralDomain

from UnivariatePolynomialCategory R

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

from CancellationAbelianMonoid

support: % -> List NonNegativeInteger

from FreeModuleCategory(R, NonNegativeInteger)

swinnertonDyerPolynomial: (NonNegativeInteger, %) -> % if R has IntegerNumberSystem

swinnertonDyerPolynomial(n, p) returns the Swinnerton-Dyer polynomial Sn.

term?: % -> Boolean

term?(p) checks whether or not p is a one term polynomial.

termRecursive?: % -> Boolean

termRecursive?(p) checks whether or not p as one term, recursively (all scalar types).

thetaQExp: (Integer, PositiveInteger, %) -> % if R has IntegerNumberSystem

thetaQExp(r,n,p) returns the q-expansion to length n of the Jacobi theta function raised to the power r, ϑ(q)^r.

totalDegree: % -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

totalDegree: (%, List SingletonAsOrderedSet) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

totalDegreeSorted: (%, List SingletonAsOrderedSet) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

trailingCoefficient: % -> R

trailingCoefficient(p) returns the trailing coefficientof p.

truncate: (%, NonNegativeInteger) -> %

truncate(p, n) return p truncated to n term. It's the remainder of the division by x^n.

unit?: % -> Boolean

unit?(p) checks whether or not p is invertible i.e. is a unit.

unitCanonical: % -> % if R has EntireRing

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has EntireRing

from EntireRing

univariate: % -> SparseUnivariatePolynomial R

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

univariate: (%, SingletonAsOrderedSet) -> SparseUnivariatePolynomial %

from PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

unmakeSUP: SparseUnivariatePolynomial R -> %

from UnivariatePolynomialCategory R

unvectorise: Vector R -> %

from UnivariatePolynomialCategory R

variables: % -> List SingletonAsOrderedSet

from MaybeSkewPolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

vectorise: (%, NonNegativeInteger) -> Vector R

from UnivariatePolynomialCategory R

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, NonNegativeInteger)

AbelianProductCategory R

AbelianSemiGroup

additiveValuation if R has Field

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

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom R

CoercibleFrom SingletonAsOrderedSet

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if R has Comparable

ConvertibleTo InputForm if R has ConvertibleTo InputForm and SingletonAsOrderedSet has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float and SingletonAsOrderedSet has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer and SingletonAsOrderedSet has ConvertibleTo Pattern Integer

ConvertibleTo String

DifferentialExtension R

DifferentialRing

Eltable(%, %)

Eltable(Fraction %, Fraction %) if R has IntegralDomain

Eltable(R, R)

EntireRing if R has EntireRing

EuclideanDomain if R has Field

Evalable %

FiniteAbelianMonoidRing(R, NonNegativeInteger)

FreeModuleCategory(R, NonNegativeInteger)

FullyLinearlyExplicitOver R

FullyRetractableTo R

GcdDomain if R has GcdDomain

Hashable if R has Hashable

IndexedDirectProductCategory(R, NonNegativeInteger)

IndexedProductCategory(R, NonNegativeInteger)

InnerEvalable(%, %)

InnerEvalable(SingletonAsOrderedSet, %)

InnerEvalable(SingletonAsOrderedSet, R)

IntegralDomain if R has IntegralDomain

JLObjectRing

JLObjectType

JLRing

JLType

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, NonNegativeInteger, SingletonAsOrderedSet)

Module %

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid

NMCommutativeRing

NMRing

NMType

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

PartialDifferentialRing SingletonAsOrderedSet

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol

PatternMatchable Float if SingletonAsOrderedSet has PatternMatchable Float and R has PatternMatchable Float

PatternMatchable Integer if SingletonAsOrderedSet has PatternMatchable Integer and R has PatternMatchable Integer

PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet)

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

PrincipalIdealDomain if R has Field

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RetractableTo SingletonAsOrderedSet

RightModule %

RightModule Fraction Integer if R has Algebra Fraction Integer

RightModule Integer if R has LinearlyExplicitOver Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if R has StepThrough

TwoSidedRecip

UniqueFactorizationDomain if R has PolynomialFactorizationExplicit

unitsKnown

UnivariatePolynomialCategory R

VariablesCommuteWithCoefficients