# LaurentPolynomial(R, UP)ΒΆ

gpol.spad line 1 [edit on github]

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

- 0: %
from AbelianMonoid

- 1: %
from MagmaWithUnit

- *: (%, %) -> %
from LeftModule %

- *: (Integer, %) -> %
from AbelianGroup

- *: (NonNegativeInteger, %) -> %
from AbelianMonoid

- *: (PositiveInteger, %) -> %
from AbelianSemiGroup

- +: (%, %) -> %
from AbelianSemiGroup

- -: % -> %
from AbelianGroup

- -: (%, %) -> %
from AbelianGroup

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- annihilate?: (%, %) -> Boolean
from Rng

- antiCommutator: (%, %) -> %

- associates?: (%, %) -> Boolean
from EntireRing

- associator: (%, %, %) -> %
from NonAssociativeRng

- characteristic: () -> NonNegativeInteger
from NonAssociativeRing

- charthRoot: % -> Union(%, failed) if R has 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 CoercibleFrom Fraction Integer

- coerce: Integer -> %
from NonAssociativeRing

- coerce: R -> %
from CoercibleFrom R

- coerce: UP -> %
from CoercibleFrom UP

- commutator: (%, %) -> %
from NonAssociativeRng

- convert: % -> Fraction UP
from ConvertibleTo Fraction UP

- D: % -> %
from DifferentialRing

- D: (%, List Symbol) -> %
- D: (%, List Symbol, List NonNegativeInteger) -> %
- D: (%, NonNegativeInteger) -> %
from DifferentialRing

- D: (%, Symbol) -> %
- D: (%, Symbol, NonNegativeInteger) -> %
- D: (%, UP -> UP) -> %
from DifferentialExtension UP

- D: (%, UP -> UP, NonNegativeInteger) -> %
from DifferentialExtension UP

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

undocumented

- differentiate: % -> %
from DifferentialRing

- differentiate: (%, List Symbol) -> %
- differentiate: (%, List Symbol, List NonNegativeInteger) -> %
- differentiate: (%, NonNegativeInteger) -> %
from DifferentialRing

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

- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has Field
from GcdDomain

- hash: % -> SingleInteger
from SetCategory

- hashUpdate!: (HashState, %) -> HashState
from SetCategory

- latex: % -> String
from SetCategory

- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has Field
from LeftOreRing

- leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- leftPower: (%, PositiveInteger) -> %
from Magma

- leftRecip: % -> Union(%, failed)
from MagmaWithUnit

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

undocumented

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

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)

- 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

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer

CoercibleFrom Integer if R has RetractableTo Integer

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