# 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 LeftModule %

*: (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: (%, %) -> %
associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
coefficient: (%, Integer) -> R

`coefficient(x, n)` undocumented

coerce: % -> %

from Algebra %

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

from CoercibleFrom R

coerce: UP -> %

from CoercibleFrom UP

commutator: (%, %) -> %
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
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

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

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

plenaryPower: (%, PositiveInteger) -> %
principalIdeal: List % -> Record(coef: List %, generator: %) if R has Field
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
retract: % -> Integer if R has RetractableTo Integer
retract: % -> R

from RetractableTo R

retract: % -> UP

from RetractableTo UP

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(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)
trailingCoefficient: % -> R

``trailingCoefficient ``undocumented

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

CommutativeRing

CommutativeStar

DifferentialRing

EntireRing

EuclideanDomain if R has Field

GcdDomain if R has Field

IntegralDomain

LeftOreRing if R has Field

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PrincipalIdealDomain if R has Field

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

unitsKnown