# UnivariatePolynomialCategory R¶

The category of univariate polynomials over a ring `R`. No particular model is assumed - implementations can be either sparse or dense.

0: %

from AbelianMonoid

1: % if R has SemiRing

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

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

from RightModule R

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

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

-: (%, %) -> % if R has AbelianGroup or % has AbelianGroup

from AbelianGroup

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

from BasicType

^: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean if R has Ring

from Rng

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

from EntireRing

associator: (%, %, %) -> % if R has Ring
binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing
characteristic: () -> NonNegativeInteger if R has Ring
charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit or R has CharacteristicNonZero
coefficient: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
coefficient: (%, NonNegativeInteger) -> R
coefficient: (%, SingletonAsOrderedSet, 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 -> % if R has Ring or R has RetractableTo Integer
coerce: R -> %

from Algebra R

coerce: SingletonAsOrderedSet -> % if R has SemiRing
commutator: (%, %) -> % if R has Ring
composite: (%, %) -> Union(%, failed) if R has IntegralDomain

`composite(p, q)` returns `h` if `p = h(q)`, and “failed” no such `h` exists.

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

`composite(f, q)` returns `h` if `f` = `h`(`q`), and “failed” is no such `h` exists.

conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
construct: List Record(k: NonNegativeInteger, c: R) -> %
constructOrdered: List Record(k: NonNegativeInteger, c: R) -> %
content: % -> R if R has GcdDomain
content: (%, SingletonAsOrderedSet) -> % if R has GcdDomain
convert: % -> InputForm if SingletonAsOrderedSet has ConvertibleTo InputForm and R has ConvertibleTo InputForm
convert: % -> Pattern Float if R has ConvertibleTo Pattern Float and SingletonAsOrderedSet has ConvertibleTo Pattern Float and R has Ring
convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer and SingletonAsOrderedSet has ConvertibleTo Pattern Integer and R has Ring
D: % -> % if R has Ring

from DifferentialRing

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

from DifferentialRing

D: (%, R -> R) -> % if R has Ring
D: (%, R -> R, NonNegativeInteger) -> % if R has Ring
D: (%, SingletonAsOrderedSet) -> % if R has Ring
D: (%, SingletonAsOrderedSet, NonNegativeInteger) -> % if R has Ring
D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol and R has Ring
D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Ring
degree: % -> NonNegativeInteger
degree: (%, List SingletonAsOrderedSet) -> List NonNegativeInteger
degree: (%, SingletonAsOrderedSet) -> NonNegativeInteger
differentiate: % -> % if R has Ring

from DifferentialRing

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

from DifferentialRing

differentiate: (%, R -> R) -> % if R has Ring
differentiate: (%, R -> R, %) -> % if R has Ring

`differentiate(p, d, x')` extends the `R`-derivation `d` to an extension `D` in `R[x]` where `Dx` is given by `x'`, and returns `Dp`.

differentiate: (%, R -> R, NonNegativeInteger) -> % if R has Ring
differentiate: (%, SingletonAsOrderedSet) -> % if R has Ring
differentiate: (%, SingletonAsOrderedSet, NonNegativeInteger) -> % if R has Ring
differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol and R has Ring
differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol and R has Ring
discriminant: % -> R if R has CommutativeRing

`discriminant(p)` returns the discriminant of the polynomial `p`.

discriminant: (%, SingletonAsOrderedSet) -> % if R has CommutativeRing
divide: (%, %) -> Record(quotient: %, remainder: %) if R has Field

from EuclideanDomain

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

`divideExponents(p, n)` returns a new polynomial resulting from dividing all exponents of the polynomial `p` by the non negative integer `n`, or “failed” if some exponent is not exactly divisible by `n`.

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

`elt(a, b)` evaluates the fraction of univariate polynomials `a` with the distinguished variable replaced by `b`.

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

`elt(a, r)` evaluates the fraction of univariate polynomials `a` with the distinguished variable replaced by the constant `r`.

euclideanSize: % -> NonNegativeInteger if R has Field

from EuclideanDomain

eval: (%, %, %) -> % if R has SemiRing

from InnerEvalable(%, %)

eval: (%, Equation %) -> % if R has SemiRing

from Evalable %

eval: (%, List %, List %) -> % if R has SemiRing

from InnerEvalable(%, %)

eval: (%, List Equation %) -> % if R has SemiRing

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)

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

from EntireRing

exquo: (%, R) -> Union(%, failed) if R has EntireRing
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
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
fmecg: (%, NonNegativeInteger, R, %) -> % if R has Ring
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

init: % if R has StepThrough

from StepThrough

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

`integrate(p)` integrates the univariate polynomial `p` with respect to its distinguished variable.

isExpt: % -> Union(Record(var: SingletonAsOrderedSet, exponent: NonNegativeInteger), failed) if R has SemiRing
isPlus: % -> Union(List %, failed)
isTimes: % -> Union(List %, failed) if R has SemiRing
karatsubaDivide: (%, NonNegativeInteger) -> Record(quotient: %, remainder: %) if R has Ring

`karatsubaDivide(p, n)` returns the same as `monicDivide(p, monomial(1, n))`

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: NonNegativeInteger, c: R)
leftPower: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

linearExtend: (NonNegativeInteger -> R, %) -> R if R has CommutativeRing
listOfTerms: % -> List Record(k: NonNegativeInteger, c: R)
mainVariable: % -> Union(SingletonAsOrderedSet, failed)
makeSUP: % -> SparseUnivariatePolynomial R

`makeSUP(p)` converts the polynomial `p` to be of type SparseUnivariatePolynomial over the same coefficients.

map: (R -> R, %) -> %
mapExponents: (NonNegativeInteger -> NonNegativeInteger, %) -> %
minimumDegree: % -> NonNegativeInteger
minimumDegree: (%, List SingletonAsOrderedSet) -> List NonNegativeInteger
minimumDegree: (%, SingletonAsOrderedSet) -> NonNegativeInteger
monicDivide: (%, %) -> Record(quotient: %, remainder: %) if R has Ring

`monicDivide(p, q)` divide the polynomial `p` by the monic polynomial `q`, returning the pair `[quotient, remainder]`. Error: if `q` isn`'t` monic.

monicDivide: (%, %, SingletonAsOrderedSet) -> Record(quotient: %, remainder: %) if R has Ring
monomial?: % -> Boolean
monomial: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
monomial: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
monomial: (R, NonNegativeInteger) -> %
monomials: % -> List %
multiEuclidean: (List %, %) -> Union(List %, failed) if R has Field

from EuclideanDomain

multiplyExponents: (%, NonNegativeInteger) -> %

`multiplyExponents(p, n)` returns a new polynomial resulting from multiplying all exponents of the polynomial `p` by the non negative integer `n`.

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

from StepThrough

numberOfMonomials: % -> NonNegativeInteger
one?: % -> Boolean if R has SemiRing

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

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

`order(p, q)` returns the largest `n` such that `q^n` divides polynomial `p` i.e. the order of `p(x)` at `q(x)=0`.

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float and R has Ring and SingletonAsOrderedSet has PatternMatchable Float
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer and R has Ring and SingletonAsOrderedSet has PatternMatchable Integer
plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing or R has Algebra Fraction Integer
pomopo!: (%, R, NonNegativeInteger, %) -> %
prime?: % -> Boolean if R has PolynomialFactorizationExplicit
primitiveMonomials: % -> List % if R has SemiRing
primitivePart: % -> % if R has GcdDomain
primitivePart: (%, SingletonAsOrderedSet) -> % if R has GcdDomain
principalIdeal: List % -> Record(coef: List %, generator: %) if R has Field
pseudoDivide: (%, %) -> Record(coef: R, quotient: %, remainder: %) if R has IntegralDomain

`pseudoDivide(p, q)` returns `[c, s, r]`, when `p' := p*lc(q)^(deg p - deg q + 1) = c * p` is pseudo right-divided by `q`, i.e. `p' = s q + r`.

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

`pseudoQuotient(p, q)` returns `s`, the quotient when `p' := p*lc(q)^(deg p - deg q + 1)` is pseudo right-divided by `q`, i.e. `p' = s q + r`.

pseudoRemainder: (%, %) -> % if R has Ring

`pseudoRemainder(p, q)` = `r`, for polynomials `p` and `q`, returns the remainder `r` when `p' := p*lc(q)^(deg p - deg q + 1)` is pseudo right-divided by `q`, i.e. `p' = s q + r`.

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

from EuclideanDomain

recip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer and R has Ring
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R) if R has Ring

from LinearlyExplicitOver R

reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer and R has Ring
reducedSystem: Matrix % -> Matrix R if R has Ring

from LinearlyExplicitOver R

reductum: % -> %
rem: (%, %) -> % if R has Field

from EuclideanDomain

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

`resultant(p, q)` returns the resultant of the polynomials `p` and `q`.

resultant: (%, %, SingletonAsOrderedSet) -> % 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: % -> SingletonAsOrderedSet if R has SemiRing
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(SingletonAsOrderedSet, failed) if R has SemiRing
rightPower: (%, NonNegativeInteger) -> % if R has SemiRing

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed) if R has SemiRing

from MagmaWithUnit

sample: %

from AbelianMonoid

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

`separate(p, q)` returns `[a, b]` such that `p = a b`, `a` is relatively prime to `q` and `b` divides some power of `q`.

shiftLeft: (%, NonNegativeInteger) -> %

`shiftLeft(p, n)` returns `p * monomial(1, n)`

shiftRight: (%, NonNegativeInteger) -> % if R has Ring

`shiftRight(p, n)` returns `monicDivide(p, monomial(1, n)).quotient`

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
squareFree: % -> Factored % if R has GcdDomain
squareFreePart: % -> % if R has GcdDomain
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
subResultantGcd: (%, %) -> % if R has IntegralDomain

`subResultantGcd(p, q)` computes the `gcd` of the polynomials `p` and `q` using the SubResultant `GCD` algorithm.

subtractIfCan: (%, %) -> Union(%, failed)
support: % -> List NonNegativeInteger
totalDegree: % -> NonNegativeInteger
totalDegree: (%, List SingletonAsOrderedSet) -> NonNegativeInteger
totalDegreeSorted: (%, List SingletonAsOrderedSet) -> 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: (%, SingletonAsOrderedSet) -> SparseUnivariatePolynomial %
unmakeSUP: SparseUnivariatePolynomial R -> %

`unmakeSUP(sup)` converts `sup` of type SparseUnivariatePolynomial(R) to be a member of the given type. Note: converse of makeSUP.

unvectorise: Vector R -> %

`unvectorise(v)` returns the polynomial which has for coefficients the entries of `v` in the increasing order.

variables: % -> List SingletonAsOrderedSet
vectorise: (%, NonNegativeInteger) -> Vector R

`vectorise(p, n)` returns `[a0, ..., a(n-1)]` where `p = a0 + a1*x + ... + a(n-1)*x^(n-1)` + higher order terms. The degree of polynomial `p` can be different from `n-1`.

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup if R has 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

DifferentialExtension R if R has Ring

DifferentialRing if R has Ring

Eltable(%, %)

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

Eltable(R, R)

EntireRing if R has EntireRing

EuclideanDomain if R has Field

Evalable % if R has SemiRing

FullyLinearlyExplicitOver R if R has Ring

GcdDomain if R has GcdDomain

Hashable if R has Hashable

InnerEvalable(%, %) if R has SemiRing

IntegralDomain if R has IntegralDomain

LeftOreRing if R has GcdDomain

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer and R has Ring

LinearlyExplicitOver R if R has Ring

Magma

MagmaWithUnit if R has SemiRing

Module % if R has CommutativeRing

Module R if R has CommutativeRing

Monoid if R has SemiRing

NonAssociativeAlgebra % if R has CommutativeRing

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has SemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol and R has Ring

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

PrincipalIdealDomain if R has Field

RightModule Integer if R has LinearlyExplicitOver Integer and R has Ring

Ring if R has Ring

Rng if R has Ring

SemiGroup

SemiRing if R has SemiRing

SemiRng

SetCategory

StepThrough if R has StepThrough

TwoSidedRecip if R has CommutativeRing

unitsKnown if R has Ring

VariablesCommuteWithCoefficients