# MultivariatePolynomial(vl, R)ΒΆ

This type is the basic representation of sparse recursive multivariate polynomials whose variables are from a user specified list of symbols. The ordering is specified by the position of the variable in the list. The coefficient ring may be non commutative, but the variables are assumed to commute.

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, IndexedExponents OrderedVariableList vl)
=: (%, %) -> 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, IndexedExponents OrderedVariableList vl)
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero or % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
from CharacteristicNonZero
coefficient: (%, IndexedExponents OrderedVariableList vl) -> R
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
coefficient: (%, List OrderedVariableList vl, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
coefficient: (%, OrderedVariableList vl, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
coefficients: % -> List R
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
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
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: OrderedVariableList vl -> %
from RetractableTo OrderedVariableList vl
coerce: R -> %
from Algebra R
commutator: (%, %) -> %
from NonAssociativeRng
conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
content: % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
content: (%, OrderedVariableList vl) -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
convert: % -> InputForm if R has ConvertibleTo InputForm
from ConvertibleTo InputForm
convert: % -> Pattern Float if R has ConvertibleTo Pattern Float
from ConvertibleTo Pattern Float
convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer
from ConvertibleTo Pattern Integer
D: (%, List OrderedVariableList vl) -> %
from PartialDifferentialRing OrderedVariableList vl
D: (%, List OrderedVariableList vl, List NonNegativeInteger) -> %
from PartialDifferentialRing OrderedVariableList vl
D: (%, OrderedVariableList vl) -> %
from PartialDifferentialRing OrderedVariableList vl
D: (%, OrderedVariableList vl, NonNegativeInteger) -> %
from PartialDifferentialRing OrderedVariableList vl
degree: % -> IndexedExponents OrderedVariableList vl
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
degree: (%, List OrderedVariableList vl) -> List NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
degree: (%, OrderedVariableList vl) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
differentiate: (%, List OrderedVariableList vl) -> %
from PartialDifferentialRing OrderedVariableList vl
differentiate: (%, List OrderedVariableList vl, List NonNegativeInteger) -> %
from PartialDifferentialRing OrderedVariableList vl
differentiate: (%, OrderedVariableList vl) -> %
from PartialDifferentialRing OrderedVariableList vl
differentiate: (%, OrderedVariableList vl, NonNegativeInteger) -> %
from PartialDifferentialRing OrderedVariableList vl
discriminant: (%, OrderedVariableList vl) -> % if R has CommutativeRing
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
eval: (%, %, %) -> %
from InnerEvalable(%, %)
eval: (%, Equation %) -> %
from Evalable %
eval: (%, List %, List %) -> %
from InnerEvalable(%, %)
eval: (%, List Equation %) -> %
from Evalable %
eval: (%, List OrderedVariableList vl, List %) -> %
from InnerEvalable(OrderedVariableList vl, %)
eval: (%, List OrderedVariableList vl, List R) -> %
from InnerEvalable(OrderedVariableList vl, R)
eval: (%, OrderedVariableList vl, %) -> %
from InnerEvalable(OrderedVariableList vl, %)
eval: (%, OrderedVariableList vl, R) -> %
from InnerEvalable(OrderedVariableList vl, R)
exquo: (%, %) -> Union(%, failed) if R has EntireRing
from EntireRing
exquo: (%, R) -> Union(%, failed) if R has EntireRing
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
factor: % -> Factored % if R has PolynomialFactorizationExplicit
from UniqueFactorizationDomain
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
fmecg: (%, IndexedExponents OrderedVariableList vl, R, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
gcd: (%, %) -> % if R has GcdDomain
from GcdDomain
gcd: List % -> % if R has GcdDomain
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has GcdDomain
from PolynomialFactorizationExplicit
ground: % -> R
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
ground?: % -> Boolean
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
isExpt: % -> Union(Record(var: OrderedVariableList vl, exponent: NonNegativeInteger), failed)
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
isPlus: % -> Union(List %, failed)
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
isTimes: % -> Union(List %, failed)
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
latex: % -> String
from SetCategory
lcm: (%, %) -> % if R has GcdDomain
from GcdDomain
lcm: List % -> % if R has GcdDomain
from GcdDomain
lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has GcdDomain
from LeftOreRing
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mainVariable: % -> Union(OrderedVariableList vl, failed)
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
map: (R -> R, %) -> %
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
mapExponents: (IndexedExponents OrderedVariableList vl -> IndexedExponents OrderedVariableList vl, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
minimumDegree: % -> IndexedExponents OrderedVariableList vl
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
minimumDegree: (%, List OrderedVariableList vl) -> List NonNegativeInteger
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
minimumDegree: (%, OrderedVariableList vl) -> NonNegativeInteger
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
monicDivide: (%, %, OrderedVariableList vl) -> Record(quotient: %, remainder: %)
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
monomial: (%, List OrderedVariableList vl, List NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
monomial: (%, OrderedVariableList vl, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
monomial: (R, IndexedExponents OrderedVariableList vl) -> %
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
monomial?: % -> Boolean
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
monomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
multivariate: (SparseUnivariatePolynomial %, OrderedVariableList vl) -> %
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
multivariate: (SparseUnivariatePolynomial R, OrderedVariableList vl) -> %
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if OrderedVariableList vl has PatternMatchable Float and R has PatternMatchable Float
from PatternMatchable Float
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if OrderedVariableList vl has PatternMatchable Integer and R has PatternMatchable Integer
from PatternMatchable Integer
pomopo!: (%, R, IndexedExponents OrderedVariableList vl, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
prime?: % -> Boolean if R has PolynomialFactorizationExplicit
from UniqueFactorizationDomain
primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
primitivePart: % -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
primitivePart: (%, OrderedVariableList vl) -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
recip: % -> Union(%, failed)
from MagmaWithUnit
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R)
from LinearlyExplicitOver R
reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer
from LinearlyExplicitOver Integer
reducedSystem: Matrix % -> Matrix R
from LinearlyExplicitOver R
reductum: % -> %
from AbelianMonoidRing(R, IndexedExponents OrderedVariableList vl)
resultant: (%, %, OrderedVariableList vl) -> % if R has CommutativeRing
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer
retract: % -> OrderedVariableList vl
from RetractableTo OrderedVariableList vl
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(OrderedVariableList vl, failed)
from RetractableTo OrderedVariableList vl
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
solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
squareFree: % -> Factored % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
squareFreePart: % -> % if R has GcdDomain
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
totalDegree: % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
totalDegree: (%, List OrderedVariableList vl) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
totalDegreeSorted: (%, List OrderedVariableList vl) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
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
univariate: % -> SparseUnivariatePolynomial R
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
univariate: (%, OrderedVariableList vl) -> SparseUnivariatePolynomial %
from PolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
variables: % -> List OrderedVariableList vl
from MaybeSkewPolynomialCategory(R, IndexedExponents OrderedVariableList vl, OrderedVariableList vl)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra % if R has CommutativeRing

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Comparable

EntireRing if R has EntireRing

GcdDomain if R has GcdDomain

InnerEvalable(%, %)

IntegralDomain if R has IntegralDomain

LeftOreRing if R has GcdDomain

Magma

MagmaWithUnit

Module % if R has CommutativeRing

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has EntireRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown

VariablesCommuteWithCoefficients