MaybeSkewPolynomialCategory(R, E, VarSet)ΒΆ

polycat.spad line 183

The category for general multi-variate possibly skew polynomials over a ring R, in variables from VarSet, with exponents from the OrderedAbelianMonoidSup.

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
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)
coefficient: (%, List VarSet, List NonNegativeInteger) -> %
coefficient(p, lv, ln) views the polynomial p as a polynomial in the variables of lv and returns the coefficient of the term lv^ln, i.e. prod(lv_i ^ ln_i).
coefficient: (%, VarSet, NonNegativeInteger) -> %
coefficient(p, v, n) views the polynomial p as a univariate polynomial in v and returns the coefficient of the v^n term.
coefficients: % -> List R
from FiniteAbelianMonoidRing(R, E)
coerce: % -> % if R has IntegralDomain and % has VariablesCommuteWithCoefficients
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
content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, E)
degree: % -> E
from AbelianMonoidRing(R, E)
degree: (%, List VarSet) -> List NonNegativeInteger
degree(p, lv) gives the list of degrees of polynomial p with respect to each of the variables in the list lv.
degree: (%, VarSet) -> NonNegativeInteger
degree(p, v) gives the degree of polynomial p with respect to the variable v.
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: % -> R
from FiniteAbelianMonoidRing(R, E)
ground?: % -> Boolean
from FiniteAbelianMonoidRing(R, E)
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
mainVariable: % -> Union(VarSet, failed)
mainVariable(p) returns the biggest variable which actually occurs in the polynomial p, or “failed” if no variables are present. fails precisely if polynomial satisfies ground?
map: (R -> R, %) -> %
from AbelianMonoidRing(R, E)
mapExponents: (E -> E, %) -> %
from FiniteAbelianMonoidRing(R, E)
minimumDegree: % -> E
from FiniteAbelianMonoidRing(R, E)
monomial: (%, List VarSet, List NonNegativeInteger) -> %
monomial(a, [v1..vn], [e1..en]) returns a*prod(vi^ei).
monomial: (%, VarSet, NonNegativeInteger) -> %
monomial(a, x, n) creates the monomial a*x^n where a is a polynomial, x is a variable and n is a nonnegative integer.
monomial: (R, E) -> %
from AbelianMonoidRing(R, E)
monomial?: % -> Boolean
from AbelianMonoidRing(R, E)
monomials: % -> List %
monomials(p) returns the list of non-zero monomials of polynomial p, i.e. monomials(sum(a_(i) X^(i))) = [a_(1) X^(1), ..., a_(n) X^(n)].
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, E)
one?: % -> Boolean if R has SemiRing
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
pomopo!: (%, R, E, %) -> %
from FiniteAbelianMonoidRing(R, E)
primitiveMonomials: % -> List % if R has SemiRing
primitiveMonomials(p) gives the list of monomials of the polynomial p with their coefficients removed. Note: primitiveMonomials(sum(a_(i) X^(i))) = [X^(1), ..., X^(n)].
primitivePart: % -> % if R has GcdDomain
from FiniteAbelianMonoidRing(R, E)
recip: % -> Union(%, failed) if R has SemiRing
from MagmaWithUnit
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has Ring and R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R) if R has Ring
from LinearlyExplicitOver R
reducedSystem: Matrix % -> Matrix Integer if R has Ring and R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix R if R has Ring
from LinearlyExplicitOver R
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
smaller?: (%, %) -> Boolean if R has Comparable
from Comparable
subtractIfCan: (%, %) -> Union(%, failed) if R has CancellationAbelianMonoid
from CancellationAbelianMonoid
totalDegree: % -> NonNegativeInteger
totalDegree(p) returns the largest sum over all monomials of all exponents of a monomial.
totalDegree: (%, List VarSet) -> NonNegativeInteger
totalDegree(p, lv) returns the maximum sum (over all monomials of polynomial p) of the variables in the list lv.
totalDegreeSorted: (%, List VarSet) -> NonNegativeInteger
totalDegreeSorted(p, lv) returns the maximum sum (over all monomials of polynomial p) of the degree in variables in the list lv. lv is assumed to be sorted in decreasing order.
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
variables: % -> List VarSet
variables(p) returns the list of those variables actually appearing in the polynomial p.
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

canonicalUnitNormal if R has canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

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

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

Comparable if R has Comparable

EntireRing if R has EntireRing

FiniteAbelianMonoidRing(R, E)

FullyLinearlyExplicitOver R if R has Ring

FullyRetractableTo R

IntegralDomain if R has IntegralDomain and % has VariablesCommuteWithCoefficients

LeftModule %

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftModule R

LinearlyExplicitOver Integer if R has Ring and R has LinearlyExplicitOver Integer

LinearlyExplicitOver R if R has Ring

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