PolynomialRing(R, E)ΒΆ

poly.spad line 317 [edit on github]

This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring), and terms indexed by their exponents (from an arbitrary ordered abelian monoid). This type is used, for example, by the DistributedMultivariatePolynomial domain where the exponent domain is a direct product of non negative integers.

0: %

from AbelianMonoid

1: % if R has SemiRing

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

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

from RightModule Fraction Integer

*: (%, R) -> %

from RightModule R

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

from LeftModule Fraction Integer

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

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

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

from AbelianGroup

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

from AbelianGroup

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

from AbelianMonoidRing(R, E)

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

from NonAssociativeSemiRng

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

from EntireRing

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

from NonAssociativeRng

binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing

from FiniteAbelianMonoidRing(R, E)

characteristic: () -> NonNegativeInteger if R has Ring

from NonAssociativeRing

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

from CharacteristicNonZero

coefficient: (%, E) -> R

from AbelianMonoidRing(R, E)

coefficients: % -> List R

from FreeModuleCategory(R, E)

coerce: % -> % if R has CommutativeRing

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

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

from Algebra Fraction Integer

coerce: Integer -> % if R has RetractableTo Integer or R has Ring

from NonAssociativeRing

coerce: R -> %

from Algebra R

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

from NonAssociativeRng

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

from IndexedProductCategory(R, E)

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

from IndexedProductCategory(R, E)

content: % -> R if R has GcdDomain

from FiniteAbelianMonoidRing(R, E)

degree: % -> E

from AbelianMonoidRing(R, E)

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

from EntireRing

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

from FiniteAbelianMonoidRing(R, E)

fmecg: (%, E, R, %) -> % if R has Ring

from FiniteAbelianMonoidRing(R, E)

ground?: % -> Boolean

from FiniteAbelianMonoidRing(R, E)

ground: % -> R

from FiniteAbelianMonoidRing(R, E)

hash: % -> SingleInteger if E has Hashable and R has Hashable

from Hashable

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

from Hashable

latex: % -> String

from SetCategory

leadingCoefficient: % -> R

from IndexedProductCategory(R, E)

leadingMonomial: % -> %

from IndexedProductCategory(R, E)

leadingSupport: % -> E

from IndexedProductCategory(R, E)

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

from IndexedProductCategory(R, E)

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

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

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

from FreeModuleCategory(R, E)

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

from IndexedDirectProductCategory(R, E)

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

from IndexedProductCategory(R, E)

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

from FiniteAbelianMonoidRing(R, E)

minimumDegree: % -> E

from FiniteAbelianMonoidRing(R, E)

monomial?: % -> Boolean

from IndexedProductCategory(R, E)

monomial: (R, E) -> %

from IndexedProductCategory(R, E)

monomials: % -> List %

from FreeModuleCategory(R, E)

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(R, E)

one?: % -> Boolean if R has SemiRing

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing or R has Algebra Fraction Integer

from NonAssociativeAlgebra %

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

from FiniteAbelianMonoidRing(R, E)

primitivePart: % -> % if R has GcdDomain

from FiniteAbelianMonoidRing(R, E)

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

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(R, E)

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

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

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

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

sample: %

from AbelianMonoid

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

from Comparable

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

from CancellationAbelianMonoid

support: % -> List E

from FreeModuleCategory(R, E)

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

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid

AbelianMonoidRing(R, E)

AbelianProductCategory R

AbelianSemiGroup

Algebra % if R has CommutativeRing

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

CoercibleTo OutputForm

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Comparable

EntireRing if R has EntireRing

FiniteAbelianMonoidRing(R, E)

FreeModuleCategory(R, E)

FullyRetractableTo R

Hashable if E has Hashable and R has Hashable

IndexedDirectProductCategory(R, E)

IndexedProductCategory(R, E)

IntegralDomain if R has IntegralDomain

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

Magma

MagmaWithUnit if R has SemiRing

Module % if R has CommutativeRing

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid if R has SemiRing

NonAssociativeAlgebra % if R has CommutativeRing

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

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

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RightModule %

RightModule Fraction Integer if R has Algebra Fraction Integer

RightModule R

Ring if R has Ring

Rng if R has Ring

SemiGroup

SemiRing if R has SemiRing

SemiRng

SetCategory

TwoSidedRecip if R has CommutativeRing

unitsKnown if R has Ring

VariablesCommuteWithCoefficients