# LaurentPolynomial(R, UP)ΒΆ

Univariate polynomials with negative and positive exponents. Author: Manuel Bronstein Date Created: May 1988

- 0: %
- from AbelianMonoid
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from Magma
- *: (Integer, %) -> %
- from AbelianGroup
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup
- =: (%, %) -> 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
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
- from CharacteristicNonZero

- coefficient: (%, Integer) -> R
`coefficient(x, n)`

undocumented- coerce: % -> %
- from Algebra %
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer
- from RetractableTo Fraction Integer
- coerce: Integer -> %
- from NonAssociativeRing
- coerce: R -> %
- from RetractableTo R
- coerce: UP -> %
- from RetractableTo UP
- commutator: (%, %) -> %
- from NonAssociativeRng
- convert: % -> Fraction UP
- from ConvertibleTo Fraction UP
- D: % -> %
- from DifferentialRing
- D: (%, List Symbol) -> %
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> %
- from PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> %
- from DifferentialRing
- D: (%, Symbol) -> %
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> %
- from PartialDifferentialRing Symbol
- D: (%, UP -> UP) -> %
- from DifferentialExtension UP
- D: (%, UP -> UP, NonNegativeInteger) -> %
- from DifferentialExtension UP

- degree: % -> Integer
`degree(x)`

undocumented- differentiate: % -> %
- from DifferentialRing
- differentiate: (%, List Symbol) -> %
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> %
- from PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> %
- from DifferentialRing
- differentiate: (%, Symbol) -> %
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> %
- from PartialDifferentialRing Symbol
- differentiate: (%, UP -> UP) -> %
- from DifferentialExtension UP
- differentiate: (%, UP -> UP, NonNegativeInteger) -> %
- from DifferentialExtension UP
- divide: (%, %) -> Record(quotient: %, remainder: %) if R has Field
- from EuclideanDomain
- euclideanSize: % -> NonNegativeInteger if R has Field
- from EuclideanDomain
- expressIdealMember: (List %, %) -> Union(List %, failed) if R has Field
- from PrincipalIdealDomain
- exquo: (%, %) -> Union(%, failed)
- from EntireRing
- extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has Field
- from EuclideanDomain
- extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has Field
- from EuclideanDomain
- gcd: (%, %) -> % if R has Field
- from GcdDomain
- gcd: List % -> % if R has Field
- from GcdDomain
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has Field
- from GcdDomain
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- latex: % -> String
- from SetCategory
- lcm: (%, %) -> % if R has Field
- from GcdDomain
- lcm: List % -> % if R has Field
- from GcdDomain
- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has Field
- from LeftOreRing

- leadingCoefficient: % -> R
- ``leadingCoefficient ``undocumented
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit

- monomial: (R, Integer) -> %
`monomial(x, n)`

undocumented

- monomial?: % -> Boolean
`monomial?(x)`

undocumented- multiEuclidean: (List %, %) -> Union(List %, failed) if R has Field
- from EuclideanDomain
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid

- order: % -> Integer
`order(x)`

undocumented- principalIdeal: List % -> Record(coef: List %, generator: %) if R has Field
- from PrincipalIdealDomain
- quo: (%, %) -> % if R has Field
- from EuclideanDomain
- recip: % -> Union(%, failed)
- from MagmaWithUnit

- reductum: % -> %
`reductum(x)`

undocumented- rem: (%, %) -> % if R has Field
- from EuclideanDomain
- 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: % -> UP
- from RetractableTo UP
- 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(UP, failed)
- from RetractableTo UP
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid

- separate: Fraction UP -> Record(polyPart: %, fracPart: Fraction UP) if R has Field
`separate(x)`

undocumented- sizeLess?: (%, %) -> Boolean if R has Field
- from EuclideanDomain
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid

- trailingCoefficient: % -> R
- ``trailingCoefficient ``undocumented
- unit?: % -> Boolean
- from EntireRing
- unitCanonical: % -> %
- from EntireRing
- unitNormal: % -> Record(unit: %, canonical: %, associate: %)
- from EntireRing
- zero?: % -> Boolean
- from AbelianMonoid

Algebra %

BiModule(%, %)

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

EuclideanDomain if R has Field

LeftOreRing if R has Field

Module %

PartialDifferentialRing Symbol

PrincipalIdealDomain if R has Field

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer