UnivariateLaurentSeriesConstructorCategory(Coef, UTS)

laurent.spad line 1 [edit on github]

This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair [n, f(x)], where n is an arbitrary integer and f(x) is a Taylor series. This pair represents the Laurent series x^n * f(x).

* : (%, %) -> %
from Magma
* : (%, Coef) -> %
from RightModule(Coef)
* : (%, UTS) -> % if Coef has Field
from RightModule(UTS)
* : (%, Fraction(Integer)) -> % if Coef has Algebra(Fraction(Integer))
from RightModule(Fraction(Integer))
* : (%, Integer) -> % if UTS has LinearlyExplicitOver(Integer) and Coef has Field
from RightModule(Integer)
* : (Coef, %) -> %
from LeftModule(Coef)
* : (UTS, %) -> % if Coef has Field
from LeftModule(UTS)
* : (Fraction(Integer), %) -> % if Coef has Algebra(Fraction(Integer))
from LeftModule(Fraction(Integer))
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
/ : (%, %) -> % if Coef has Field
from Field
/ : (%, Coef) -> % if Coef has Field
from AbelianMonoidRing(Coef, Integer)
/ : (UTS, UTS) -> % if Coef has Field
from QuotientFieldCategory(UTS)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
< : (%, %) -> Boolean if UTS has OrderedSet and Coef has Field
from PartialOrder
<= : (%, %) -> Boolean if UTS has OrderedSet and Coef has Field
from PartialOrder
= : (%, %) -> Boolean
from BasicType
> : (%, %) -> Boolean if UTS has OrderedSet and Coef has Field
from PartialOrder
>= : (%, %) -> Boolean if UTS has OrderedSet and Coef has Field
from PartialOrder
D : % -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
D : (%, List(Symbol)) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, Mapping(UTS, UTS)) -> % if Coef has Field
from DifferentialExtension(UTS)
D : (%, Mapping(UTS, UTS), NonNegativeInteger) -> % if Coef has Field
from DifferentialExtension(UTS)
D : (%, NonNegativeInteger) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
D : (%, Symbol) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, Symbol, NonNegativeInteger) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
^ : (%, %) -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
^ : (%, Fraction(Integer)) -> % if Coef has Algebra(Fraction(Integer))
from RadicalCategory
^ : (%, Integer) -> % if Coef has Field
from DivisionRing
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
abs : % -> % if Coef has Field and UTS has OrderedIntegralDomain
from OrderedRing
acos : % -> % if Coef has Algebra(Fraction(Integer))
from ArcTrigonometricFunctionCategory
acosh : % -> % if Coef has Algebra(Fraction(Integer))
from ArcHyperbolicFunctionCategory
acot : % -> % if Coef has Algebra(Fraction(Integer))
from ArcTrigonometricFunctionCategory
acoth : % -> % if Coef has Algebra(Fraction(Integer))
from ArcHyperbolicFunctionCategory
acsc : % -> % if Coef has Algebra(Fraction(Integer))
from ArcTrigonometricFunctionCategory
acsch : % -> % if Coef has Algebra(Fraction(Integer))
from ArcHyperbolicFunctionCategory
annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
approximate : (%, Integer) -> Coef if Coef has coerce : Symbol -> Coef and Coef has ^ : (Coef, Integer) -> Coef
from UnivariatePowerSeriesCategory(Coef, Integer)
asec : % -> % if Coef has Algebra(Fraction(Integer))
from ArcTrigonometricFunctionCategory
asech : % -> % if Coef has Algebra(Fraction(Integer))
from ArcHyperbolicFunctionCategory
asin : % -> % if Coef has Algebra(Fraction(Integer))
from ArcTrigonometricFunctionCategory
asinh : % -> % if Coef has Algebra(Fraction(Integer))
from ArcHyperbolicFunctionCategory
associates? : (%, %) -> Boolean if Coef has IntegralDomain
from EntireRing
associator : (%, %, %) -> %
from NonAssociativeRng
atan : % -> % if Coef has Algebra(Fraction(Integer))
from ArcTrigonometricFunctionCategory
atanh : % -> % if Coef has Algebra(Fraction(Integer))
from ArcHyperbolicFunctionCategory
ceiling : % -> UTS if UTS has IntegerNumberSystem and Coef has Field
from QuotientFieldCategory(UTS)
center : % -> Coef
from UnivariatePowerSeriesCategory(Coef, Integer)
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
charthRoot : % -> Union(%, "failed") if Coef has CharacteristicNonZero or Coef has Field
from CharacteristicNonZero
coefficient : (%, Integer) -> Coef
from AbelianMonoidRing(Coef, Integer)
coerce : % -> % if Coef has CommutativeRing
from Algebra(%)
coerce : Coef -> % if Coef has CommutativeRing
from Algebra(Coef)
coerce : UTS -> %

coerce(f(x)) converts the Taylor series f(x) to a Laurent series.

coerce : Fraction(Integer) -> % if Coef has Algebra(Fraction(Integer))
from Algebra(Fraction(Integer))
coerce : Integer -> %
from NonAssociativeRing
coerce : Symbol -> % if UTS has RetractableTo(Symbol) and Coef has Field
from CoercibleFrom(Symbol)
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
complete : % -> %
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
conditionP : Matrix(%) -> Union(Vector(%), "failed") if % has CharacteristicNonZero and Coef has Field and UTS has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
construct : List(Record(k : Integer, c : Coef)) -> %
from IndexedProductCategory(Coef, Integer)
constructOrdered : List(Record(k : Integer, c : Coef)) -> %
from IndexedProductCategory(Coef, Integer)
convert : % -> DoubleFloat if UTS has RealConstant and Coef has Field
from ConvertibleTo(DoubleFloat)
convert : % -> Float if UTS has RealConstant and Coef has Field
from ConvertibleTo(Float)
convert : % -> InputForm if Coef has Field and UTS has ConvertibleTo(InputForm)
from ConvertibleTo(InputForm)
convert : % -> Pattern(Float) if Coef has Field and UTS has ConvertibleTo(Pattern(Float))
from ConvertibleTo(Pattern(Float))
convert : % -> Pattern(Integer) if Coef has Field and UTS has ConvertibleTo(Pattern(Integer))
from ConvertibleTo(Pattern(Integer))
cos : % -> % if Coef has Algebra(Fraction(Integer))
from TrigonometricFunctionCategory
cosh : % -> % if Coef has Algebra(Fraction(Integer))
from HyperbolicFunctionCategory
cot : % -> % if Coef has Algebra(Fraction(Integer))
from TrigonometricFunctionCategory
coth : % -> % if Coef has Algebra(Fraction(Integer))
from HyperbolicFunctionCategory
csc : % -> % if Coef has Algebra(Fraction(Integer))
from TrigonometricFunctionCategory
csch : % -> % if Coef has Algebra(Fraction(Integer))
from HyperbolicFunctionCategory
degree : % -> Integer

degree(f(x)) returns the degree of the lowest order term of f(x), which may have zero as a coefficient.

denom : % -> UTS if Coef has Field
from QuotientFieldCategory(UTS)
denominator : % -> % if Coef has Field
from QuotientFieldCategory(UTS)
differentiate : % -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
differentiate : (%, List(Symbol)) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, Mapping(UTS, UTS)) -> % if Coef has Field
from DifferentialExtension(UTS)
differentiate : (%, Mapping(UTS, UTS), NonNegativeInteger) -> % if Coef has Field
from DifferentialExtension(UTS)
differentiate : (%, NonNegativeInteger) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
differentiate : (%, Symbol) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, Symbol, NonNegativeInteger) -> % if Coef has Field or Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
divide : (%, %) -> Record(quotient : %, remainder : %) if Coef has Field
from EuclideanDomain
elt : (%, %) -> %
from Eltable(%, %)
elt : (%, UTS) -> % if UTS has Eltable(UTS, UTS) and Coef has Field
from Eltable(UTS, %)
elt : (%, Integer) -> Coef
from UnivariatePowerSeriesCategory(Coef, Integer)
euclideanSize : % -> NonNegativeInteger if Coef has Field
from EuclideanDomain
eval : (%, UTS, UTS) -> % if UTS has Evalable(UTS) and Coef has Field
from InnerEvalable(UTS, UTS)
eval : (%, Equation(UTS)) -> % if UTS has Evalable(UTS) and Coef has Field
from Evalable(UTS)
eval : (%, List(UTS), List(UTS)) -> % if UTS has Evalable(UTS) and Coef has Field
from InnerEvalable(UTS, UTS)
eval : (%, List(Equation(UTS))) -> % if UTS has Evalable(UTS) and Coef has Field
from Evalable(UTS)
eval : (%, List(Symbol), List(UTS)) -> % if UTS has InnerEvalable(Symbol, UTS) and Coef has Field
from InnerEvalable(Symbol, UTS)
eval : (%, Symbol, UTS) -> % if UTS has InnerEvalable(Symbol, UTS) and Coef has Field
from InnerEvalable(Symbol, UTS)
eval : (%, Coef) -> Stream(Coef) if Coef has ^ : (Coef, Integer) -> Coef
from UnivariatePowerSeriesCategory(Coef, Integer)
exp : % -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
expressIdealMember : (List(%), %) -> Union(List(%), "failed") if Coef has Field
from PrincipalIdealDomain
exquo : (%, %) -> Union(%, "failed") if Coef has IntegralDomain
from EntireRing
extend : (%, Integer) -> %
from UnivariatePowerSeriesCategory(Coef, Integer)
extendedEuclidean : (%, %) -> Record(coef1 : %, coef2 : %, generator : %) if Coef has Field
from EuclideanDomain
extendedEuclidean : (%, %, %) -> Union(Record(coef1 : %, coef2 : %), "failed") if Coef has Field
from EuclideanDomain
factor : % -> Factored(%) if Coef has Field
from UniqueFactorizationDomain
factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Coef has Field and UTS has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Coef has Field and UTS has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
floor : % -> UTS if UTS has IntegerNumberSystem and Coef has Field
from QuotientFieldCategory(UTS)
fractionPart : % -> % if UTS has EuclideanDomain and Coef has Field
from QuotientFieldCategory(UTS)
gcd : (%, %) -> % if Coef has Field
from GcdDomain
gcd : List(%) -> % if Coef has Field
from GcdDomain
gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if Coef has Field
from GcdDomain
init : () -> % if UTS has StepThrough and Coef has Field
from StepThrough
integrate : % -> % if Coef has Algebra(Fraction(Integer))
from UnivariateSeriesWithRationalExponents(Coef, Integer)
integrate : (%, Symbol) -> % if Coef has Algebra(Fraction(Integer)) and Coef has integrate : (Coef, Symbol) -> Coef and Coef has variables : Coef -> List(Symbol)
from UnivariateSeriesWithRationalExponents(Coef, Integer)
inv : % -> % if Coef has Field
from DivisionRing
latex : % -> String
from SetCategory
laurent : (Integer, UTS) -> %

laurent(n, f(x)) returns x^n * f(x).

laurent : (Integer, Stream(Coef)) -> %
from UnivariateLaurentSeriesCategory(Coef)
lcm : (%, %) -> % if Coef has Field
from GcdDomain
lcm : List(%) -> % if Coef has Field
from GcdDomain
lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %) if Coef has Field
from LeftOreRing
leadingCoefficient : % -> Coef
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
leadingMonomial : % -> %
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
leadingSupport : % -> Integer
from IndexedProductCategory(Coef, Integer)
leadingTerm : % -> Record(k : Integer, c : Coef)
from IndexedProductCategory(Coef, Integer)
leftPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower : (%, PositiveInteger) -> %
from Magma
leftRecip : % -> Union(%, "failed")
from MagmaWithUnit
log : % -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
map : (Mapping(Coef, Coef), %) -> %
from IndexedProductCategory(Coef, Integer)
map : (Mapping(UTS, UTS), %) -> % if Coef has Field
from FullyEvalableOver(UTS)
max : (%, %) -> % if UTS has OrderedSet and Coef has Field
from OrderedSet
min : (%, %) -> % if UTS has OrderedSet and Coef has Field
from OrderedSet
monomial : (Coef, Integer) -> %
from IndexedProductCategory(Coef, Integer)
monomial? : % -> Boolean
from IndexedProductCategory(Coef, Integer)
multiEuclidean : (List(%), %) -> Union(List(%), "failed") if Coef has Field
from EuclideanDomain
multiplyCoefficients : (Mapping(Coef, Integer), %) -> %
from UnivariateLaurentSeriesCategory(Coef)
multiplyExponents : (%, PositiveInteger) -> %
from UnivariatePowerSeriesCategory(Coef, Integer)
negative? : % -> Boolean if Coef has Field and UTS has OrderedIntegralDomain
from OrderedRing
nextItem : % -> Union(%, "failed") if UTS has StepThrough and Coef has Field
from StepThrough
nthRoot : (%, Integer) -> % if Coef has Algebra(Fraction(Integer))
from RadicalCategory
numer : % -> UTS if Coef has Field
from QuotientFieldCategory(UTS)
numerator : % -> % if Coef has Field
from QuotientFieldCategory(UTS)
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
order : % -> Integer
from UnivariatePowerSeriesCategory(Coef, Integer)
order : (%, Integer) -> Integer
from UnivariatePowerSeriesCategory(Coef, Integer)
patternMatch : (%, Pattern(Float), PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if UTS has PatternMatchable(Float) and Coef has Field
from PatternMatchable(Float)
patternMatch : (%, Pattern(Integer), PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if UTS has PatternMatchable(Integer) and Coef has Field
from PatternMatchable(Integer)
pi : () -> % if Coef has Algebra(Fraction(Integer))
from TranscendentalFunctionCategory
plenaryPower : (%, PositiveInteger) -> % if Coef has CommutativeRing or Coef has Algebra(Fraction(Integer))
from NonAssociativeAlgebra(Coef)
pole? : % -> Boolean
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
positive? : % -> Boolean if Coef has Field and UTS has OrderedIntegralDomain
from OrderedRing
prime? : % -> Boolean if Coef has Field
from UniqueFactorizationDomain
principalIdeal : List(%) -> Record(coef : List(%), generator : %) if Coef has Field
from PrincipalIdealDomain
quo : (%, %) -> % if Coef has Field
from EuclideanDomain
rationalFunction : (%, Integer) -> Fraction(Polynomial(Coef)) if Coef has IntegralDomain
from UnivariateLaurentSeriesCategory(Coef)
rationalFunction : (%, Integer, Integer) -> Fraction(Polynomial(Coef)) if Coef has IntegralDomain
from UnivariateLaurentSeriesCategory(Coef)
recip : % -> Union(%, "failed")
from MagmaWithUnit
reducedSystem : Matrix(%) -> Matrix(UTS) if Coef has Field
from LinearlyExplicitOver(UTS)
reducedSystem : Matrix(%) -> Matrix(Integer) if UTS has LinearlyExplicitOver(Integer) and Coef has Field
from LinearlyExplicitOver(Integer)
reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(UTS), vec : Vector(UTS)) if Coef has Field
from LinearlyExplicitOver(UTS)
reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Integer), vec : Vector(Integer)) if UTS has LinearlyExplicitOver(Integer) and Coef has Field
from LinearlyExplicitOver(Integer)
reductum : % -> %
from IndexedProductCategory(Coef, Integer)
rem : (%, %) -> % if Coef has Field
from EuclideanDomain
removeZeroes : % -> %

removeZeroes(f(x)) removes leading zeroes from the representation of the Laurent series f(x). A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient, the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable. Note: removeZeroes(f) removes all leading zeroes from f

removeZeroes : (Integer, %) -> %

removeZeroes(n, f(x)) removes up to n leading zeroes from the Laurent series f(x). A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient, the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable.

retract : % -> UTS
from RetractableTo(UTS)
retract : % -> Fraction(Integer) if UTS has RetractableTo(Integer) and Coef has Field
from RetractableTo(Fraction(Integer))
retract : % -> Integer if UTS has RetractableTo(Integer) and Coef has Field
from RetractableTo(Integer)
retract : % -> Symbol if UTS has RetractableTo(Symbol) and Coef has Field
from RetractableTo(Symbol)
retractIfCan : % -> Union(UTS, "failed")
from RetractableTo(UTS)
retractIfCan : % -> Union(Fraction(Integer), "failed") if UTS has RetractableTo(Integer) and Coef has Field
from RetractableTo(Fraction(Integer))
retractIfCan : % -> Union(Integer, "failed") if UTS has RetractableTo(Integer) and Coef has Field
from RetractableTo(Integer)
retractIfCan : % -> Union(Symbol, "failed") if UTS has RetractableTo(Symbol) and Coef has Field
from RetractableTo(Symbol)
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
sample : () -> %
from AbelianMonoid
sec : % -> % if Coef has Algebra(Fraction(Integer))
from TrigonometricFunctionCategory
sech : % -> % if Coef has Algebra(Fraction(Integer))
from HyperbolicFunctionCategory
series : Stream(Record(k : Integer, c : Coef)) -> %
from UnivariateLaurentSeriesCategory(Coef)
sign : % -> Integer if Coef has Field and UTS has OrderedIntegralDomain
from OrderedRing
sin : % -> % if Coef has Algebra(Fraction(Integer))
from TrigonometricFunctionCategory
sinh : % -> % if Coef has Algebra(Fraction(Integer))
from HyperbolicFunctionCategory
sizeLess? : (%, %) -> Boolean if Coef has Field
from EuclideanDomain
smaller? : (%, %) -> Boolean if UTS has Comparable and Coef has Field
from Comparable
solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)), SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)), "failed") if Coef has Field and UTS has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
sqrt : % -> % if Coef has Algebra(Fraction(Integer))
from RadicalCategory
squareFree : % -> Factored(%) if Coef has Field
from UniqueFactorizationDomain
squareFreePart : % -> % if Coef has Field
from UniqueFactorizationDomain
squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%)) if Coef has Field and UTS has PolynomialFactorizationExplicit
from PolynomialFactorizationExplicit
subtractIfCan : (%, %) -> Union(%, "failed")
from CancellationAbelianMonoid
tan : % -> % if Coef has Algebra(Fraction(Integer))
from TrigonometricFunctionCategory
tanh : % -> % if Coef has Algebra(Fraction(Integer))
from HyperbolicFunctionCategory
taylor : % -> UTS

taylor(f(x)) converts the Laurent series f(x) to a Taylor series, if possible. Error: if this is not possible.

taylorIfCan : % -> Union(UTS, "failed")

taylorIfCan(f(x)) converts the Laurent series f(x) to a Taylor series, if possible. If this is not possible, "failed" is returned.

taylorRep : % -> UTS

taylorRep(f(x)) returns g(x), where f = x^n * g(x) is represented by [n, g(x)].

terms : % -> Stream(Record(k : Integer, c : Coef))
from UnivariatePowerSeriesCategory(Coef, Integer)
truncate : (%, Integer) -> %
from UnivariatePowerSeriesCategory(Coef, Integer)
truncate : (%, Integer, Integer) -> %
from UnivariatePowerSeriesCategory(Coef, Integer)
unit? : % -> Boolean if Coef has IntegralDomain
from EntireRing
unitCanonical : % -> % if Coef has IntegralDomain
from EntireRing
unitNormal : % -> Record(unit : %, canonical : %, associate : %) if Coef has IntegralDomain
from EntireRing
variable : % -> Symbol
from UnivariatePowerSeriesCategory(Coef, Integer)
wholePart : % -> UTS if UTS has EuclideanDomain and Coef has Field
from QuotientFieldCategory(UTS)
zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

EntireRing

Ring

Algebra(%)

BiModule(UTS, UTS)

ConvertibleTo(Float)

Eltable(%, %)

CancellationAbelianMonoid

CommutativeStar

FullyLinearlyExplicitOver(UTS)

InnerEvalable(Symbol, UTS)

ConvertibleTo(Pattern(Integer))

Field

UnivariateSeriesWithRationalExponents(Coef, Integer)

LinearlyExplicitOver(Integer)

QuotientFieldCategory(UTS)

SemiGroup

OrderedSet

UnivariateLaurentSeriesCategory(Coef)

LeftModule(%)

VariablesCommuteWithCoefficients

ConvertibleTo(DoubleFloat)

ArcTrigonometricFunctionCategory

LeftModule(UTS)

PartialOrder

UnivariatePowerSeriesCategory(Coef, Integer)

NonAssociativeAlgebra(Coef)

Monoid

OrderedAbelianSemiGroup

BiModule(Fraction(Integer), Fraction(Integer))

TrigonometricFunctionCategory

Module(%)

Eltable(UTS, %)

TranscendentalFunctionCategory

LeftOreRing

NonAssociativeSemiRing

noZeroDivisors

ConvertibleTo(InputForm)

Rng

SemiRing

NonAssociativeAlgebra(%)

SetCategory

CharacteristicNonZero

IndexedProductCategory(Coef, Integer)

PolynomialFactorizationExplicit

TwoSidedRecip

EuclideanDomain

CoercibleFrom(Fraction(Integer))

NonAssociativeRing

RealConstant

CoercibleFrom(Symbol)

InnerEvalable(UTS, UTS)

Module(UTS)

Patternable(UTS)

NonAssociativeRng

Module(Coef)

RetractableTo(Integer)

Algebra(Coef)

Comparable

PrincipalIdealDomain

canonicalsClosed

DivisionRing

LinearlyExplicitOver(UTS)

BiModule(Coef, Coef)

ArcHyperbolicFunctionCategory

ElementaryFunctionCategory

RetractableTo(UTS)

unitsKnown

AbelianSemiGroup

RadicalCategory

canonicalUnitNormal

LeftModule(Fraction(Integer))

PatternMatchable(Float)

Evalable(UTS)

AbelianProductCategory(Coef)

IntegralDomain

RightModule(Integer)

NonAssociativeAlgebra(Fraction(Integer))

NonAssociativeSemiRng

NonAssociativeAlgebra(UTS)

GcdDomain

CharacteristicZero

CommutativeRing

CoercibleFrom(UTS)

Algebra(Fraction(Integer))

HyperbolicFunctionCategory

AbelianGroup

OrderedRing

DifferentialRing

BasicType

DifferentialExtension(UTS)

RetractableTo(Fraction(Integer))

MagmaWithUnit

CoercibleFrom(Integer)

AbelianMonoid

PatternMatchable(Integer)

CoercibleTo(OutputForm)

FullyPatternMatchable(UTS)

BiModule(%, %)

LeftModule(Coef)

StepThrough

OrderedCancellationAbelianMonoid

SemiRng

RetractableTo(Symbol)

RightModule(Coef)

RightModule(Fraction(Integer))

OrderedAbelianMonoid

PartialDifferentialRing(Symbol)

OrderedAbelianGroup

FullyEvalableOver(UTS)

Algebra(UTS)

UniqueFactorizationDomain

RightModule(UTS)

OrderedIntegralDomain

ConvertibleTo(Pattern(Float))

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

Module(Fraction(Integer))

AbelianMonoidRing(Coef, Integer)

RightModule(%)

Magma