PolynomialRing(R, E)ΒΆ
poly.spad line 317 [edit on github]
R: Join(SemiRng, AbelianMonoid)
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 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, %) -> % 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)
- ^: (%, NonNegativeInteger) -> % if R has SemiRing
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- annihilate?: (%, %) -> Boolean if R has Ring
from Rng
- antiCommutator: (%, %) -> %
- 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
- 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
- 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
- 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)
- 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
AbelianMonoidRing(R, E)
Algebra % if R has CommutativeRing
Algebra Fraction Integer if R has Algebra Fraction Integer
Algebra R if R has CommutativeRing
BiModule(%, %)
BiModule(Fraction Integer, Fraction Integer) if R has Algebra Fraction Integer
BiModule(R, R)
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
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)
Hashable if E has Hashable and R has Hashable
IndexedDirectProductCategory(R, E)
IndexedProductCategory(R, E)
IntegralDomain if R has IntegralDomain
LeftModule Fraction Integer if R has Algebra Fraction Integer
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
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
noZeroDivisors if R has EntireRing
RetractableTo Fraction Integer if R has RetractableTo Fraction Integer
RetractableTo Integer if R has RetractableTo Integer
RightModule Fraction Integer if R has Algebra Fraction Integer
TwoSidedRecip if R has CommutativeRing
unitsKnown if R has Ring