# PolynomialCategory(R, E, VarSet)ΒΆ

- R: Join(SemiRng, AbelianMonoid)
- E: OrderedAbelianMonoidSup
- VarSet: OrderedSet

The category for general multi-variate polynomials over a ring `R`

, in variables from VarSet, with exponents from the OrderedAbelianMonoidSup. Here variables commute with the coefficients.

- 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) -> %
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- coefficient: (%, VarSet, NonNegativeInteger) -> %
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- coefficients: % -> List R
- from FiniteAbelianMonoidRing(R, E)
- coerce: % -> % if R has IntegralDomain
- 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
- coerce: VarSet -> % if R has SemiRing
- from RetractableTo VarSet
- commutator: (%, %) -> % if R has Ring
- from NonAssociativeRng
- content: % -> R if R has GcdDomain
- from FiniteAbelianMonoidRing(R, E)

- content: (%, VarSet) -> % if R has GcdDomain
`content(p, v)`

is the`gcd`

of the coefficients of the polynomial`p`

when`p`

is viewed as a univariate polynomial with respect to the variable`v`

. Thus, for polynomial 7*x^2*y + 14*x*y^2, the`gcd`

of the coefficients with respect to`x`

is 7*y.- convert: % -> InputForm if VarSet has ConvertibleTo InputForm and R has ConvertibleTo InputForm
- from ConvertibleTo InputForm
- convert: % -> Pattern Float if R has Ring and VarSet has ConvertibleTo Pattern Float and R has ConvertibleTo Pattern Float
- from ConvertibleTo Pattern Float
- convert: % -> Pattern Integer if R has Ring and VarSet has ConvertibleTo Pattern Integer and R has ConvertibleTo Pattern Integer
- from ConvertibleTo Pattern Integer
- D: (%, List VarSet) -> % if R has Ring
- from PartialDifferentialRing VarSet
- D: (%, List VarSet, List NonNegativeInteger) -> % if R has Ring
- from PartialDifferentialRing VarSet
- D: (%, VarSet) -> % if R has Ring
- from PartialDifferentialRing VarSet
- D: (%, VarSet, NonNegativeInteger) -> % if R has Ring
- from PartialDifferentialRing VarSet
- degree: % -> E
- from AbelianMonoidRing(R, E)
- degree: (%, List VarSet) -> List NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- degree: (%, VarSet) -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- differentiate: (%, List VarSet) -> % if R has Ring
- from PartialDifferentialRing VarSet
- differentiate: (%, List VarSet, List NonNegativeInteger) -> % if R has Ring
- from PartialDifferentialRing VarSet
- differentiate: (%, VarSet) -> % if R has Ring
- from PartialDifferentialRing VarSet
- differentiate: (%, VarSet, NonNegativeInteger) -> % if R has Ring
- from PartialDifferentialRing VarSet

- discriminant: (%, VarSet) -> % if R has CommutativeRing
`discriminant(p, v)`

returns the disriminant of the polynomial`p`

with respect to the variable`v`

.- eval: (%, %, %) -> % if R has SemiRing
- from InnerEvalable(%, %)
- eval: (%, Equation %) -> % if R has SemiRing
- from Evalable %
- eval: (%, List %, List %) -> % if R has SemiRing
- from InnerEvalable(%, %)
- eval: (%, List Equation %) -> % if R has SemiRing
- from Evalable %
- eval: (%, List VarSet, List %) -> %
- from InnerEvalable(VarSet, %)
- eval: (%, List VarSet, List R) -> %
- from InnerEvalable(VarSet, R)
- eval: (%, VarSet, %) -> %
- from InnerEvalable(VarSet, %)
- eval: (%, VarSet, R) -> %
- from InnerEvalable(VarSet, R)
- exquo: (%, %) -> Union(%, failed) if R has EntireRing
- from EntireRing
- exquo: (%, R) -> Union(%, failed) if R has EntireRing
- from FiniteAbelianMonoidRing(R, E)
- 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: (%, E, R, %) -> % if R has Ring
- from FiniteAbelianMonoidRing(R, E)
- 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, E)
- ground?: % -> Boolean
- from FiniteAbelianMonoidRing(R, E)
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory

- isExpt: % -> Union(Record(var: VarSet, exponent: NonNegativeInteger), failed) if R has SemiRing
`isExpt(p)`

returns`[x, n]`

if polynomial`p`

has the form`x^n`

and`n > 0`

.

- isPlus: % -> Union(List %, failed)
`isPlus(p)`

returns`[m1, ..., mn]`

if polynomial`p = m1 + ... + mn`

and`n >= 2`

and each`mi`

is a nonzero monomial.

- isTimes: % -> Union(List %, failed) if R has SemiRing
`isTimes(p)`

returns`[a1, ..., an]`

if polynomial`p = a1 ... an`

and`n >= 2`

, and, for each`i`

,`ai`

is either a nontrivial constant in`R`

or else of the form`x^e`

, where`e > 0`

is an integer and`x`

is a member of VarSet.- 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
- 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)
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- map: (R -> R, %) -> %
- from AbelianMonoidRing(R, E)
- mapExponents: (E -> E, %) -> %
- from FiniteAbelianMonoidRing(R, E)
- minimumDegree: % -> E
- from FiniteAbelianMonoidRing(R, E)

- minimumDegree: (%, List VarSet) -> List NonNegativeInteger
`minimumDegree(p, lv)`

gives the list of minimum degrees of the polynomial`p`

with respect to each of the variables in the list`lv`

- minimumDegree: (%, VarSet) -> NonNegativeInteger
`minimumDegree(p, v)`

gives the minimum degree of polynomial`p`

with respect to`v`

, i.e. viewed a univariate polynomial in`v`

- monicDivide: (%, %, VarSet) -> Record(quotient: %, remainder: %) if R has Ring
`monicDivide(a, b, v)`

divides the polynomial a by the polynomial`b`

, with each viewed as a univariate polynomial in`v`

returning both the quotient and remainder. Error: if`b`

is not monic with respect to`v`

.- monomial: (%, List VarSet, List NonNegativeInteger) -> %
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- monomial: (%, VarSet, NonNegativeInteger) -> %
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- monomial: (R, E) -> %
- from AbelianMonoidRing(R, E)
- monomial?: % -> Boolean
- from AbelianMonoidRing(R, E)
- monomials: % -> List %
- from MaybeSkewPolynomialCategory(R, E, VarSet)

- multivariate: (SparseUnivariatePolynomial %, VarSet) -> %
`multivariate(sup, v)`

converts an anonymous univariable polynomial`sup`

to a polynomial in the variable`v`

.

- multivariate: (SparseUnivariatePolynomial R, VarSet) -> %
`multivariate(sup, v)`

converts an anonymous univariable polynomial`sup`

to a polynomial in the variable`v`

.- numberOfMonomials: % -> NonNegativeInteger
- from FiniteAbelianMonoidRing(R, E)
- one?: % -> Boolean if R has SemiRing
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has Ring and R has PatternMatchable Float and VarSet has PatternMatchable Float
- from PatternMatchable Float
- patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has Ring and R has PatternMatchable Integer and VarSet has PatternMatchable Integer
- from PatternMatchable Integer
- pomopo!: (%, R, E, %) -> %
- from FiniteAbelianMonoidRing(R, E)
- prime?: % -> Boolean if R has PolynomialFactorizationExplicit
- from UniqueFactorizationDomain
- primitiveMonomials: % -> List % if R has SemiRing
- from MaybeSkewPolynomialCategory(R, E, VarSet)

- primitivePart: % -> % if R has GcdDomain
`primitivePart(p)`

returns the unitCanonical associate of the polynomial`p`

with its content divided out.

- primitivePart: (%, VarSet) -> % if R has GcdDomain
`primitivePart(p, v)`

returns the unitCanonical associate of the polynomial`p`

with its content with respect to the variable`v`

divided out.- 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)

- resultant: (%, %, VarSet) -> % if R has CommutativeRing
`resultant(p, q, v)`

returns the resultant of the polynomials`p`

and`q`

with respect to the variable`v`

.- 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
- retract: % -> VarSet if R has SemiRing
- from RetractableTo VarSet
- 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
- retractIfCan: % -> Union(VarSet, failed) if R has SemiRing
- from RetractableTo VarSet
- 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
- solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if R has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit

- squareFree: % -> Factored % if R has GcdDomain
`squareFree(p)`

returns the square free factorization of the polynomial`p`

.

- squareFreePart: % -> % if R has GcdDomain
`squareFreePart(p)`

returns product of all the irreducible factors of polynomial`p`

each taken with multiplicity one.- squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if R has PolynomialFactorizationExplicit
- from PolynomialFactorizationExplicit
- subtractIfCan: (%, %) -> Union(%, failed) if R has CancellationAbelianMonoid
- from CancellationAbelianMonoid
- totalDegree: % -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- totalDegree: (%, List VarSet) -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- totalDegreeSorted: (%, List VarSet) -> NonNegativeInteger
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- 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
`univariate(p)`

converts the multivariate polynomial`p`

, which should actually involve only one variable, into a univariate polynomial in that variable, whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate

- univariate: (%, VarSet) -> SparseUnivariatePolynomial %
`univariate(p, v)`

converts the multivariate polynomial`p`

into a univariate polynomial in`v`

, whose coefficients are still multivariate polynomials (in all the other variables).- variables: % -> List VarSet
- from MaybeSkewPolynomialCategory(R, E, VarSet)
- zero?: % -> Boolean
- from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoidRing(R, E)

Algebra % if R has IntegralDomain

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)

CancellationAbelianMonoid if R has CancellationAbelianMonoid

canonicalUnitNormal if R has canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Comparable

ConvertibleTo InputForm if VarSet has ConvertibleTo InputForm and R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has Ring and VarSet has ConvertibleTo Pattern Float and R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has Ring and VarSet has ConvertibleTo Pattern Integer and R has ConvertibleTo Pattern Integer

EntireRing if R has EntireRing

FiniteAbelianMonoidRing(R, E)

FullyLinearlyExplicitOver R if R has Ring

InnerEvalable(%, %) if R has SemiRing

InnerEvalable(VarSet, %)

InnerEvalable(VarSet, R)

IntegralDomain if R has IntegralDomain

LeftModule Fraction Integer if R has Algebra Fraction Integer

LeftOreRing if R has GcdDomain

LinearlyExplicitOver Integer if R has Ring and R has LinearlyExplicitOver Integer

LinearlyExplicitOver R if R has Ring

MagmaWithUnit if R has SemiRing

MaybeSkewPolynomialCategory(R, E, VarSet)

Module % if R has IntegralDomain

Module Fraction Integer if R has Algebra Fraction Integer

Module 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

PartialDifferentialRing VarSet if R has Ring

PatternMatchable Float if R has Ring and R has PatternMatchable Float and VarSet has PatternMatchable Float

PatternMatchable Integer if R has Ring and R has PatternMatchable Integer and VarSet has PatternMatchable Integer

PolynomialFactorizationExplicit if R has PolynomialFactorizationExplicit

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo VarSet if R has SemiRing

RightModule Fraction Integer if R has Algebra Fraction Integer

UniqueFactorizationDomain if R has PolynomialFactorizationExplicit

unitsKnown if R has Ring