SymmetricPolynomial(R)

prtition.spad line 125 [edit on github]

This domain implements symmetric polynomial

* : (%, %) -> %
from Magma
* : (%, R) -> %
from RightModule(R)
* : (%, Fraction(Integer)) -> % if R has Algebra(Fraction(Integer))
from RightModule(Fraction(Integer))
* : (R, %) -> %
from LeftModule(R)
* : (Fraction(Integer), %) -> % if R has Algebra(Fraction(Integer))
from LeftModule(Fraction(Integer))
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
/ : (%, R) -> % if R has Field
from AbelianMonoidRing(R, Partition)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
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 FreeModuleCategory(R, Partition)
coefficients : % -> List(R)
from FreeModuleCategory(R, Partition)
coerce : % -> % if R has CommutativeRing
from Algebra(%)
coerce : R -> %
from Algebra(R)
coerce : Fraction(Integer) -> % if R has Algebra(Fraction(Integer)) or R has RetractableTo(Fraction(Integer))
from Algebra(Fraction(Integer))
coerce : Integer -> %
from NonAssociativeRing
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
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 : % -> R
from FiniteAbelianMonoidRing(R, Partition)
ground? : % -> Boolean
from FiniteAbelianMonoidRing(R, Partition)
hash : % -> SingleInteger if Partition has Hashable and R has Hashable
from Hashable
hashUpdate! : (HashState, %) -> HashState if Partition has Hashable and R 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 : (Mapping(R, Partition), %) -> R if R has CommutativeRing
from FreeModuleCategory(R, Partition)
listOfTerms : % -> List(Record(k : Partition, c : R))
from IndexedDirectProductCategory(R, Partition)
map : (Mapping(R, R), %) -> %
from IndexedProductCategory(R, Partition)
mapExponents : (Mapping(Partition, Partition), %) -> %
from FiniteAbelianMonoidRing(R, Partition)
minimumDegree : % -> Partition
from FiniteAbelianMonoidRing(R, Partition)
monomial : (R, Partition) -> %
from IndexedProductCategory(R, Partition)
monomial? : % -> Boolean
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 CommutativeRing or R has Algebra(Fraction(Integer))
from NonAssociativeAlgebra(%)
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 : % -> R
from RetractableTo(R)
retract : % -> Fraction(Integer) if R has RetractableTo(Fraction(Integer))
from RetractableTo(Fraction(Integer))
retract : % -> Integer if R has RetractableTo(Integer)
from RetractableTo(Integer)
retractIfCan : % -> Union(R, "failed")
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)
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
~= : (%, %) -> Boolean
from BasicType

CharacteristicNonZero

Module(Fraction(Integer))

Comparable

LeftModule(Fraction(Integer))

CoercibleFrom(R)

noZeroDivisors

RightModule(%)

IndexedProductCategory(R, Partition)

Algebra(R)

Monoid

AbelianMonoid

Algebra(%)

BiModule(R, R)

NonAssociativeAlgebra(Fraction(Integer))

CancellationAbelianMonoid

AbelianMonoidRing(R, Partition)

MagmaWithUnit

FiniteAbelianMonoidRing(R, Partition)

RightModule(R)

RightModule(Fraction(Integer))

RetractableTo(Integer)

AbelianSemiGroup

NonAssociativeSemiRng

LeftModule(%)

LeftModule(R)

canonicalUnitNormal

Module(%)

SetCategory

CoercibleTo(OutputForm)

Algebra(Fraction(Integer))

Rng

CommutativeRing

IntegralDomain

TwoSidedRecip

Magma

NonAssociativeAlgebra(R)

CoercibleFrom(Fraction(Integer))

FreeModuleCategory(R, Partition)

SemiGroup

CoercibleFrom(Integer)

AbelianGroup

RetractableTo(Fraction(Integer))

CommutativeStar

NonAssociativeSemiRing

AbelianProductCategory(R)

VariablesCommuteWithCoefficients

NonAssociativeAlgebra(%)

Module(R)

BiModule(Fraction(Integer), Fraction(Integer))

IndexedDirectProductCategory(R, Partition)

CharacteristicZero

RetractableTo(R)

NonAssociativeRng

unitsKnown

Ring

NonAssociativeRing

SemiRng

EntireRing

Hashable

BasicType

BiModule(%, %)

SemiRing

FullyRetractableTo(R)