SymmetricPolynomial R

prtition.spad line 125 [edit on github]

• R: Ring

This domain implements symmetric polynomial

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, 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, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

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

from AbelianMonoidRing(R, Partition)

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

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

from EntireRing

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

from NonAssociativeRng

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

from FiniteAbelianMonoidRing(R, Partition)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

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

from CharacteristicNonZero

coefficient: (%, Partition) -> R

from AbelianMonoidRing(R, Partition)

coefficients: % -> List R

from FreeModuleCategory(R, Partition)

coerce: % -> % if R has CommutativeRing

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

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

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: R -> %

from CoercibleFrom R

commutator: (%, %) -> %

from NonAssociativeRng

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

from IndexedProductCategory(R, Partition)

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

from IndexedProductCategory(R, Partition)

content: % -> R if R has GcdDomain

from FiniteAbelianMonoidRing(R, Partition)

degree: % -> Partition

from AbelianMonoidRing(R, Partition)

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

from EntireRing

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

from FiniteAbelianMonoidRing(R, Partition)

fmecg: (%, Partition, R, %) -> %

from FiniteAbelianMonoidRing(R, Partition)

ground?: % -> Boolean

from FiniteAbelianMonoidRing(R, Partition)

ground: % -> R

from FiniteAbelianMonoidRing(R, Partition)

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

from Hashable

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

from Hashable

latex: % -> String

from SetCategory

leadingCoefficient: % -> R

from IndexedProductCategory(R, Partition)

leadingMonomial: % -> %

from IndexedProductCategory(R, Partition)

leadingSupport: % -> Partition

from IndexedProductCategory(R, Partition)

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

from IndexedProductCategory(R, Partition)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

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

from FreeModuleCategory(R, Partition)

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

from IndexedDirectProductCategory(R, Partition)

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

from IndexedProductCategory(R, Partition)

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

from FiniteAbelianMonoidRing(R, Partition)

minimumDegree: % -> Partition

from FiniteAbelianMonoidRing(R, Partition)

monomial?: % -> Boolean

from IndexedProductCategory(R, Partition)

monomial: (R, Partition) -> %

from IndexedProductCategory(R, Partition)

monomials: % -> List %

from FreeModuleCategory(R, Partition)

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(R, Partition)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

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

from NonAssociativeAlgebra Fraction Integer

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

from FiniteAbelianMonoidRing(R, Partition)

primitivePart: % -> % if R has GcdDomain

from FiniteAbelianMonoidRing(R, Partition)

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(R, Partition)

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) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

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

from Comparable

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

from CancellationAbelianMonoid

support: % -> List Partition

from FreeModuleCategory(R, Partition)

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

AbelianMonoid

AbelianMonoidRing(R, Partition)

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, Partition)

FreeModuleCategory(R, Partition)

FullyRetractableTo R

Hashable if R has Hashable and Partition has Hashable

IndexedDirectProductCategory(R, Partition)

IndexedProductCategory(R, Partition)

IntegralDomain if R has IntegralDomain

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

Magma

MagmaWithUnit

Module % if R has CommutativeRing

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing

Monoid

NonAssociativeAlgebra % if R has CommutativeRing

NonAssociativeAlgebra Fraction Integer if R has Algebra Fraction Integer

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

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

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if R has CommutativeRing

unitsKnown

VariablesCommuteWithCoefficients