AlgebraicallyClosedFunctionSpace R

algfunc.spad line 147 [edit on github]

• R: Join(Comparable, IntegralDomain)

Model for algebraically closed function spaces.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (%, Integer) -> % if R has LinearlyExplicitOver Integer

from RightModule Integer

*: (%, R) -> %

from RightModule R

*: (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

/: (%, %) -> %

from Field

/: (SparseMultivariatePolynomial(R, Kernel %), SparseMultivariatePolynomial(R, Kernel %)) -> %

from FunctionSpace2(R, Kernel %)

=: (%, %) -> Boolean

from BasicType

^: (%, Fraction Integer) -> %

from RadicalCategory

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

algtower: % -> List Kernel %

from FunctionSpace2(R, Kernel %)

algtower: List % -> List Kernel %

from FunctionSpace2(R, Kernel %)

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

applyQuote: (Symbol, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, %, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, %, %, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, %, %, %, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, List %) -> %

from FunctionSpace2(R, Kernel %)

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

belong?: BasicOperator -> Boolean

from ExpressionSpace2 Kernel %

box: % -> %

from ExpressionSpace2 Kernel %

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero

from CharacteristicNonZero

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: AlgebraicNumber -> % if R has RetractableTo Integer

from CoercibleFrom AlgebraicNumber

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Fraction Polynomial Fraction R -> %

from FunctionSpace2(R, Kernel %)

coerce: Fraction Polynomial R -> %

from CoercibleFrom Fraction Polynomial R

coerce: Fraction R -> %

from FunctionSpace2(R, Kernel %)

coerce: Integer -> %

from NonAssociativeRing

coerce: Kernel % -> %

from CoercibleFrom Kernel %

coerce: Polynomial Fraction R -> %

from FunctionSpace2(R, Kernel %)

coerce: Polynomial R -> %

from CoercibleFrom Polynomial R

coerce: R -> %

from Algebra R

coerce: SparseMultivariatePolynomial(R, Kernel %) -> %

from FunctionSpace2(R, Kernel %)

coerce: Symbol -> %

from CoercibleFrom Symbol

commutator: (%, %) -> %

from NonAssociativeRng

conjugate: (%, %) -> % if R has Group

from Group

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

convert: Factored % -> %

from FunctionSpace2(R, Kernel %)

D: (%, List Symbol) -> %

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> %

from PartialDifferentialRing Symbol

D: (%, Symbol) -> %

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> %

from PartialDifferentialRing Symbol

definingPolynomial: % -> %

from ExpressionSpace2 Kernel %

denom: % -> SparseMultivariatePolynomial(R, Kernel %)

from FunctionSpace2(R, Kernel %)

denominator: % -> %

from FunctionSpace2(R, Kernel %)

differentiate: (%, List Symbol) -> %

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> %

from PartialDifferentialRing Symbol

differentiate: (%, Symbol) -> %

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> %

from PartialDifferentialRing Symbol

distribute: % -> %

from ExpressionSpace2 Kernel %

distribute: (%, %) -> %

from ExpressionSpace2 Kernel %

divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

elt: (BasicOperator, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, List %) -> %

from ExpressionSpace2 Kernel %

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

eval: (%, %, %) -> %

from InnerEvalable(%, %)

eval: (%, BasicOperator, % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, BasicOperator, %, Symbol) -> % if R has ConvertibleTo InputForm

from FunctionSpace2(R, Kernel %)

eval: (%, BasicOperator, List % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, Equation %) -> %

from Evalable %

eval: (%, Kernel %, %) -> %

from InnerEvalable(Kernel %, %)

eval: (%, List %, List %) -> %

from InnerEvalable(%, %)

eval: (%, List BasicOperator, List %, Symbol) -> % if R has ConvertibleTo InputForm

from FunctionSpace2(R, Kernel %)

eval: (%, List BasicOperator, List(% -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, List BasicOperator, List(List % -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, List Equation %) -> %

from Evalable %

eval: (%, List Kernel %, List %) -> %

from InnerEvalable(Kernel %, %)

eval: (%, List Symbol, List NonNegativeInteger, List(% -> %)) -> %

from FunctionSpace2(R, Kernel %)

eval: (%, List Symbol, List NonNegativeInteger, List(List % -> %)) -> %

from FunctionSpace2(R, Kernel %)

eval: (%, List Symbol, List(% -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, List Symbol, List(List % -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, Symbol, % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, Symbol, List % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, Symbol, NonNegativeInteger, % -> %) -> %

from FunctionSpace2(R, Kernel %)

eval: (%, Symbol, NonNegativeInteger, List % -> %) -> %

from FunctionSpace2(R, Kernel %)

even?: % -> Boolean if % has RetractableTo Integer

from ExpressionSpace2 Kernel %

expressIdealMember: (List %, %) -> Union(List %, failed)

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed)

from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)

from EuclideanDomain

factor: % -> Factored %

from UniqueFactorizationDomain

freeOf?: (%, %) -> Boolean

from ExpressionSpace2 Kernel %

freeOf?: (%, Symbol) -> Boolean

from ExpressionSpace2 Kernel %

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from GcdDomain

ground?: % -> Boolean

from FunctionSpace2(R, Kernel %)

ground: % -> R

from FunctionSpace2(R, Kernel %)

height: % -> NonNegativeInteger

from ExpressionSpace2 Kernel %

inv: % -> %

from DivisionRing

is?: (%, BasicOperator) -> Boolean

from ExpressionSpace2 Kernel %

is?: (%, Symbol) -> Boolean

from ExpressionSpace2 Kernel %

isExpt: % -> Union(Record(var: Kernel %, exponent: Integer), failed)

from FunctionSpace2(R, Kernel %)

isExpt: (%, BasicOperator) -> Union(Record(var: Kernel %, exponent: Integer), failed)

from FunctionSpace2(R, Kernel %)

isExpt: (%, Symbol) -> Union(Record(var: Kernel %, exponent: Integer), failed)

from FunctionSpace2(R, Kernel %)

isMult: % -> Union(Record(coef: Integer, var: Kernel %), failed)

from FunctionSpace2(R, Kernel %)

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

from FunctionSpace2(R, Kernel %)

isPower: % -> Union(Record(val: %, exponent: Integer), failed)

from FunctionSpace2(R, Kernel %)

isTimes: % -> Union(List %, failed)

from FunctionSpace2(R, Kernel %)

kernel: (BasicOperator, %) -> %

from ExpressionSpace2 Kernel %

kernel: (BasicOperator, List %) -> %

from ExpressionSpace2 Kernel %

kernels: % -> List Kernel %

from ExpressionSpace2 Kernel %

kernels: List % -> List Kernel %

from ExpressionSpace2 Kernel %

latex: % -> String

from SetCategory

lcm: (%, %) -> %

from GcdDomain

lcm: List % -> %

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

mainKernel: % -> Union(Kernel %, failed)

from ExpressionSpace2 Kernel %

map: (% -> %, Kernel %) -> %

from ExpressionSpace2 Kernel %

minPoly: Kernel % -> SparseUnivariatePolynomial %

from ExpressionSpace2 Kernel %

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

nthRoot: (%, Integer) -> %

from RadicalCategory

numer: % -> SparseMultivariatePolynomial(R, Kernel %)

from FunctionSpace2(R, Kernel %)

numerator: % -> %

from FunctionSpace2(R, Kernel %)

odd?: % -> Boolean if % has RetractableTo Integer

from ExpressionSpace2 Kernel %

one?: % -> Boolean

from MagmaWithUnit

operator: BasicOperator -> BasicOperator

from ExpressionSpace2 Kernel %

operators: % -> List BasicOperator

from ExpressionSpace2 Kernel %

opposite?: (%, %) -> Boolean

from AbelianMonoid

paren: % -> %

from ExpressionSpace2 Kernel %

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra Fraction Integer

prime?: % -> Boolean

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %)

from PrincipalIdealDomain

quo: (%, %) -> %

from EuclideanDomain

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

rem: (%, %) -> %

from EuclideanDomain

retract: % -> AlgebraicNumber if R has RetractableTo Integer

from RetractableTo AlgebraicNumber

retract: % -> Fraction Integer if R has RetractableTo Integer or R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Fraction Polynomial R

from RetractableTo Fraction Polynomial R

retract: % -> Integer if R has RetractableTo Integer

from RetractableTo Integer

retract: % -> Kernel %

from RetractableTo Kernel %

retract: % -> Polynomial R

from RetractableTo Polynomial R

retract: % -> R

from RetractableTo R

retract: % -> Symbol

from RetractableTo Symbol

retractIfCan: % -> Union(AlgebraicNumber, failed) if R has RetractableTo Integer

from RetractableTo AlgebraicNumber

retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Integer or R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Fraction Polynomial R, failed)

from RetractableTo Fraction Polynomial R

retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(Kernel %, failed)

from RetractableTo Kernel %

retractIfCan: % -> Union(Polynomial R, failed)

from RetractableTo Polynomial R

retractIfCan: % -> Union(R, failed)

from RetractableTo R

retractIfCan: % -> Union(Symbol, failed)

from RetractableTo Symbol

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rootOf: % -> %

rootOf(p) returns y such that p(y) = 0. Error: if p has more than one variable y.

rootOf: (%, Symbol) -> %

rootOf(p, y) returns y such that p(y) = 0. The object returned displays as 'y.

rootOf: (SparseUnivariatePolynomial %, Symbol) -> %

from AlgebraicallyClosedField

rootOf: Polynomial % -> %

from AlgebraicallyClosedField

rootOf: SparseUnivariatePolynomial % -> %

from AlgebraicallyClosedField

rootsOf: % -> List %

rootsOf(p, y) returns [y1, ..., yn] such that p(yi) = 0; Note: the returned values y1, …, yn contain new symbols which are bound in the interpreter to the respective values. Error: if p has more than one variable y.

rootsOf: (%, Symbol) -> List %

rootsOf(p, y) returns [y1, ..., yn] such that p(yi) = 0; The returned roots contain new symbols '\%z0, '\%z1 …; Note: the new symbols are bound in the interpreter to the respective values.

rootsOf: (SparseUnivariatePolynomial %, Symbol) -> List %

from AlgebraicallyClosedField

rootsOf: Polynomial % -> List %

from AlgebraicallyClosedField

rootsOf: SparseUnivariatePolynomial % -> List %

from AlgebraicallyClosedField

rootSum: (%, SparseUnivariatePolynomial %, Symbol) -> %

sample: %

from AbelianMonoid

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

sqrt: % -> %

from RadicalCategory

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

subst: (%, Equation %) -> %

from ExpressionSpace2 Kernel %

subst: (%, List Equation %) -> %

from ExpressionSpace2 Kernel %

subst: (%, List Kernel %, List %) -> %

from ExpressionSpace2 Kernel %

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

tower: % -> List Kernel %

from ExpressionSpace2 Kernel %

tower: List % -> List Kernel %

from ExpressionSpace2 Kernel %

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

univariate: (%, Kernel %) -> Fraction SparseUnivariatePolynomial %

from FunctionSpace2(R, Kernel %)

variables: % -> List Symbol

from FunctionSpace2(R, Kernel %)

variables: List % -> List Symbol

from FunctionSpace2(R, Kernel %)

zero?: % -> Boolean

from AbelianMonoid

zeroOf: % -> %

zeroOf(p) returns y such that p(y) = 0. The value y is expressed in terms of radicals if possible, and otherwise as an implicit algebraic quantity. Error: if p has more than one variable.

zeroOf: (%, Symbol) -> %

zeroOf(p, y) returns y such that p(y) = 0. The value y is expressed in terms of radicals if possible, and otherwise as an implicit algebraic quantity which displays as 'y.

zeroOf: (SparseUnivariatePolynomial %, Symbol) -> %

from AlgebraicallyClosedField

zeroOf: Polynomial % -> %

from AlgebraicallyClosedField

zeroOf: SparseUnivariatePolynomial % -> %

from AlgebraicallyClosedField

zerosOf: % -> List %

zerosOf(p) returns [y1, ..., yn] such that p(yi) = 0. The yi's are expressed in radicals if possible. Note: the returned values y1, …, yn contain new symbols which are bound in the interpreter to the respective values. Error: if p has more than one variable.

zerosOf: (%, Symbol) -> List %

zerosOf(p, y) returns [y1, ..., yn] such that p(yi) = 0. The yi's are expressed in radicals if possible, and otherwise as implicit algebraic quantities containing new symbols which display as '\%z0, '\%z1, …; The new symbols are bound in the interpreter to the respective values.

zerosOf: (SparseUnivariatePolynomial %, Symbol) -> List %

from AlgebraicallyClosedField

zerosOf: Polynomial % -> List %

from AlgebraicallyClosedField

zerosOf: SparseUnivariatePolynomial % -> List %

from AlgebraicallyClosedField

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra R

AlgebraicallyClosedField

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(R, R)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom AlgebraicNumber if R has RetractableTo Integer

CoercibleFrom Fraction Integer if R has RetractableTo Integer or R has RetractableTo Fraction Integer

CoercibleFrom Fraction Polynomial R

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom Kernel %

CoercibleFrom Polynomial R

CoercibleFrom R

CoercibleFrom Symbol

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo InputForm if R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

DivisionRing

EntireRing

EuclideanDomain

Evalable %

ExpressionSpace

ExpressionSpace2 Kernel %

Field

FullyLinearlyExplicitOver R

FullyPatternMatchable R

FullyRetractableTo R

FunctionSpace R

FunctionSpace2(R, Kernel %)

GcdDomain

Group if R has Group

InnerEvalable(%, %)

InnerEvalable(Kernel %, %)

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule R

LeftOreRing

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer

LinearlyExplicitOver R

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module R

Monoid

NonAssociativeAlgebra %

NonAssociativeAlgebra Fraction Integer

NonAssociativeAlgebra R

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PartialDifferentialRing Symbol

Patternable R

PatternMatchable Float if R has PatternMatchable Float

PatternMatchable Integer if R has PatternMatchable Integer

PrincipalIdealDomain

RadicalCategory

RetractableTo AlgebraicNumber if R has RetractableTo Integer

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer or R has RetractableTo Integer

RetractableTo Fraction Polynomial R

RetractableTo Integer if R has RetractableTo Integer

RetractableTo Kernel %

RetractableTo Polynomial R

RetractableTo R

RetractableTo Symbol

RightModule %

RightModule Fraction Integer

RightModule Integer if R has LinearlyExplicitOver Integer

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown