FiniteAbelianMonoidRing(R, E)ΒΆ

polycat.spad line 68

This category is similar to AbelianMonoidRing, except that the sum is assumed to be finite. It is a useful model for polynomials, but is somewhat more general.

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 R has AbelianGroup
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> % if R has AbelianGroup
from AbelianGroup
-: (%, %) -> % if 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
binomThmExpt(p, q, n) returns (p+q)^n by means of the binomial theorem trick.
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
coefficients(p) gives the list of non-zero coefficients of polynomial p.
coerce: % -> % if R has IntegralDomain and % has VariablesCommuteWithCoefficients
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer or R has Algebra 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
content: % -> R if R has GcdDomain
content(p) gives the gcd of the coefficients of polynomial p.
degree: % -> E
from AbelianMonoidRing(R, E)
exquo: (%, %) -> Union(%, failed) if R has EntireRing
from EntireRing
exquo: (%, R) -> Union(%, failed) if R has EntireRing
exquo(p,r) returns the exact quotient of polynomial p by r, or “failed” if none exists.
fmecg: (%, E, R, %) -> % if R has Ring
fmecg(p1, e, r, p2) returns p1 - monomial(r, e) * p2.
ground: % -> R
ground(p) retracts polynomial p to the coefficient ring.
ground?: % -> Boolean
ground?(p) tests if polynomial p is a member of the coefficient ring.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> R
from AbelianMonoidRing(R, E)
leadingMonomial: % -> %
from AbelianMonoidRing(R, E)
leftPower: (%, NonNegativeInteger) -> % if R has SemiRing
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
map: (R -> R, %) -> %
from AbelianMonoidRing(R, E)
mapExponents: (E -> E, %) -> %
mapExponents(fn, u) maps function fn onto the exponents of the non-zero monomials of polynomial u.
minimumDegree: % -> E
minimumDegree(p) gives the least exponent of a non-zero term of polynomial p. Error: if applied to 0.
monomial: (R, E) -> %
from AbelianMonoidRing(R, E)
monomial?: % -> Boolean
from AbelianMonoidRing(R, E)
numberOfMonomials: % -> NonNegativeInteger
numberOfMonomials(p) gives the number of non-zero monomials in polynomial p.
one?: % -> Boolean if R has SemiRing
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
pomopo!: (%, R, E, %) -> %
pomopo!(p1, r, e, p2) returns p1 + monomial(r, e) * p2 and may use p1 as workspace. The constant r is assumed to be nonzero.
primitivePart: % -> % if R has GcdDomain
primitivePart(p) returns the unit normalized form of polynomial p divided by the content of p.
recip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
reductum: % -> %
from AbelianMonoidRing(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
subtractIfCan: (%, %) -> Union(%, failed) if R has CancellationAbelianMonoid
from CancellationAbelianMonoid
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)

AbelianSemiGroup

Algebra % if R has IntegralDomain and % has VariablesCommuteWithCoefficients

Algebra Fraction Integer if R has Algebra Fraction Integer

Algebra R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if R has Algebra Fraction Integer

BiModule(R, R)

CancellationAbelianMonoid if R has CancellationAbelianMonoid

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients

CommutativeStar if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has CommutativeRing and % has VariablesCommuteWithCoefficients

EntireRing if R has EntireRing

FullyRetractableTo R

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

Magma

MagmaWithUnit if R has SemiRing

Module % if R has IntegralDomain and % has VariablesCommuteWithCoefficients

Module Fraction Integer if R has Algebra Fraction Integer

Module R if R has CommutativeRing and % has VariablesCommuteWithCoefficients

Monoid if R has SemiRing

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

unitsKnown if R has Ring