BalancedPAdicRational p

padic.spad line 559 [edit on github]

• p: Integer

Stream-based implementation of Qp: numbers are represented as sum(i = k.., a[i] * p^i), where the a[i] lie in -(p - 1)/2, …, (p - 1)/2.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, BalancedPAdicInteger p) -> %

from RightModule BalancedPAdicInteger p

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

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

from RightModule Integer

*: (BalancedPAdicInteger p, %) -> %

from LeftModule BalancedPAdicInteger p

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

/: (BalancedPAdicInteger p, BalancedPAdicInteger p) -> %

from QuotientFieldCategory BalancedPAdicInteger p

<=: (%, %) -> Boolean if BalancedPAdicInteger p has OrderedSet

from PartialOrder

<: (%, %) -> Boolean if BalancedPAdicInteger p has OrderedSet

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean if BalancedPAdicInteger p has OrderedSet

from PartialOrder

>: (%, %) -> Boolean if BalancedPAdicInteger p has OrderedSet

from PartialOrder

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

abs: % -> % if BalancedPAdicInteger p has OrderedIntegralDomain

from OrderedRing

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

approximate: (%, Integer) -> Fraction Integer

associates?: (%, %) -> Boolean

from EntireRing

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

from NonAssociativeRng

ceiling: % -> BalancedPAdicInteger p if BalancedPAdicInteger p has IntegerNumberSystem

from QuotientFieldCategory BalancedPAdicInteger p

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if BalancedPAdicInteger p has CharacteristicNonZero or % has CharacteristicNonZero and BalancedPAdicInteger p has PolynomialFactorizationExplicit

from CharacteristicNonZero

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: BalancedPAdicInteger p -> %

from CoercibleFrom BalancedPAdicInteger p

coerce: Fraction Integer -> %

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: Symbol -> % if BalancedPAdicInteger p has RetractableTo Symbol

from CoercibleFrom Symbol

commutator: (%, %) -> %

from NonAssociativeRng

conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and BalancedPAdicInteger p has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

continuedFraction: % -> ContinuedFraction Fraction Integer

convert: % -> DoubleFloat if BalancedPAdicInteger p has RealConstant

from ConvertibleTo DoubleFloat

convert: % -> Float if BalancedPAdicInteger p has RealConstant

from ConvertibleTo Float

convert: % -> InputForm if BalancedPAdicInteger p has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if BalancedPAdicInteger p has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if BalancedPAdicInteger p has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

D: % -> % if BalancedPAdicInteger p has DifferentialRing

from DifferentialRing

D: (%, BalancedPAdicInteger p -> BalancedPAdicInteger p) -> %

from DifferentialExtension BalancedPAdicInteger p

D: (%, BalancedPAdicInteger p -> BalancedPAdicInteger p, NonNegativeInteger) -> %

from DifferentialExtension BalancedPAdicInteger p

D: (%, List Symbol) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if BalancedPAdicInteger p has DifferentialRing

from DifferentialRing

D: (%, Symbol) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

denom: % -> BalancedPAdicInteger p

from QuotientFieldCategory BalancedPAdicInteger p

denominator: % -> %

from QuotientFieldCategory BalancedPAdicInteger p

differentiate: % -> % if BalancedPAdicInteger p has DifferentialRing

from DifferentialRing

differentiate: (%, BalancedPAdicInteger p -> BalancedPAdicInteger p) -> %

from DifferentialExtension BalancedPAdicInteger p

differentiate: (%, BalancedPAdicInteger p -> BalancedPAdicInteger p, NonNegativeInteger) -> %

from DifferentialExtension BalancedPAdicInteger p

differentiate: (%, List Symbol) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if BalancedPAdicInteger p has DifferentialRing

from DifferentialRing

differentiate: (%, Symbol) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if BalancedPAdicInteger p has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

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

from EuclideanDomain

elt: (%, BalancedPAdicInteger p) -> % if BalancedPAdicInteger p has Eltable(BalancedPAdicInteger p, BalancedPAdicInteger p)

from Eltable(BalancedPAdicInteger p, %)

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

eval: (%, BalancedPAdicInteger p, BalancedPAdicInteger p) -> % if BalancedPAdicInteger p has Evalable BalancedPAdicInteger p

from InnerEvalable(BalancedPAdicInteger p, BalancedPAdicInteger p)

eval: (%, Equation BalancedPAdicInteger p) -> % if BalancedPAdicInteger p has Evalable BalancedPAdicInteger p

from Evalable BalancedPAdicInteger p

eval: (%, List BalancedPAdicInteger p, List BalancedPAdicInteger p) -> % if BalancedPAdicInteger p has Evalable BalancedPAdicInteger p

from InnerEvalable(BalancedPAdicInteger p, BalancedPAdicInteger p)

eval: (%, List Equation BalancedPAdicInteger p) -> % if BalancedPAdicInteger p has Evalable BalancedPAdicInteger p

from Evalable BalancedPAdicInteger p

eval: (%, List Symbol, List BalancedPAdicInteger p) -> % if BalancedPAdicInteger p has InnerEvalable(Symbol, BalancedPAdicInteger p)

from InnerEvalable(Symbol, BalancedPAdicInteger p)

eval: (%, Symbol, BalancedPAdicInteger p) -> % if BalancedPAdicInteger p has InnerEvalable(Symbol, BalancedPAdicInteger p)

from InnerEvalable(Symbol, BalancedPAdicInteger p)

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

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if BalancedPAdicInteger p has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if BalancedPAdicInteger p has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

floor: % -> BalancedPAdicInteger p if BalancedPAdicInteger p has IntegerNumberSystem

from QuotientFieldCategory BalancedPAdicInteger p

fractionPart: % -> %

from QuotientFieldCategory BalancedPAdicInteger p

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

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

from PolynomialFactorizationExplicit

init: % if BalancedPAdicInteger p has StepThrough

from StepThrough

inv: % -> %

from DivisionRing

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

map: (BalancedPAdicInteger p -> BalancedPAdicInteger p, %) -> %

from FullyEvalableOver BalancedPAdicInteger p

max: (%, %) -> % if BalancedPAdicInteger p has OrderedSet

from OrderedSet

min: (%, %) -> % if BalancedPAdicInteger p has OrderedSet

from OrderedSet

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

from EuclideanDomain

negative?: % -> Boolean if BalancedPAdicInteger p has OrderedIntegralDomain

from OrderedRing

nextItem: % -> Union(%, failed) if BalancedPAdicInteger p has StepThrough

from StepThrough

numer: % -> BalancedPAdicInteger p

from QuotientFieldCategory BalancedPAdicInteger p

numerator: % -> %

from QuotientFieldCategory BalancedPAdicInteger p

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

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

from PatternMatchable Float

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

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

positive?: % -> Boolean if BalancedPAdicInteger p has OrderedIntegralDomain

from OrderedRing

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 BalancedPAdicInteger p, vec: Vector BalancedPAdicInteger p)

from LinearlyExplicitOver BalancedPAdicInteger p

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if BalancedPAdicInteger p has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix BalancedPAdicInteger p

from LinearlyExplicitOver BalancedPAdicInteger p

reducedSystem: Matrix % -> Matrix Integer if BalancedPAdicInteger p has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

rem: (%, %) -> %

from EuclideanDomain

removeZeroes: % -> %

removeZeroes: (Integer, %) -> %

retract: % -> BalancedPAdicInteger p

from RetractableTo BalancedPAdicInteger p

retract: % -> Fraction Integer if BalancedPAdicInteger p has RetractableTo Integer

from RetractableTo Fraction Integer

retract: % -> Integer if BalancedPAdicInteger p has RetractableTo Integer

from RetractableTo Integer

retract: % -> Symbol if BalancedPAdicInteger p has RetractableTo Symbol

from RetractableTo Symbol

retractIfCan: % -> Union(BalancedPAdicInteger p, failed)

from RetractableTo BalancedPAdicInteger p

retractIfCan: % -> Union(Fraction Integer, failed) if BalancedPAdicInteger p has RetractableTo Integer

from RetractableTo Fraction Integer

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

from RetractableTo Integer

retractIfCan: % -> Union(Symbol, failed) if BalancedPAdicInteger p has RetractableTo Symbol

from RetractableTo Symbol

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sign: % -> Integer if BalancedPAdicInteger p has OrderedIntegralDomain

from OrderedRing

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean if BalancedPAdicInteger p has Comparable

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if BalancedPAdicInteger p has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if BalancedPAdicInteger p has PolynomialFactorizationExplicit

from PolynomialFactorizationExplicit

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

from CancellationAbelianMonoid

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

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

from EntireRing

wholePart: % -> BalancedPAdicInteger p

from QuotientFieldCategory BalancedPAdicInteger p

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra BalancedPAdicInteger p

Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(BalancedPAdicInteger p, BalancedPAdicInteger p)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if BalancedPAdicInteger p has CharacteristicNonZero

CharacteristicZero

CoercibleFrom BalancedPAdicInteger p

CoercibleFrom Fraction Integer if BalancedPAdicInteger p has RetractableTo Integer

CoercibleFrom Integer if BalancedPAdicInteger p has RetractableTo Integer

CoercibleFrom Symbol if BalancedPAdicInteger p has RetractableTo Symbol

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable if BalancedPAdicInteger p has Comparable

ConvertibleTo DoubleFloat if BalancedPAdicInteger p has RealConstant

ConvertibleTo Float if BalancedPAdicInteger p has RealConstant

ConvertibleTo InputForm if BalancedPAdicInteger p has ConvertibleTo InputForm

ConvertibleTo Pattern Float if BalancedPAdicInteger p has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if BalancedPAdicInteger p has ConvertibleTo Pattern Integer

DifferentialExtension BalancedPAdicInteger p

DifferentialRing if BalancedPAdicInteger p has DifferentialRing

DivisionRing

Eltable(BalancedPAdicInteger p, %) if BalancedPAdicInteger p has Eltable(BalancedPAdicInteger p, BalancedPAdicInteger p)

EntireRing

EuclideanDomain

Evalable BalancedPAdicInteger p if BalancedPAdicInteger p has Evalable BalancedPAdicInteger p

Field

FullyEvalableOver BalancedPAdicInteger p

FullyLinearlyExplicitOver BalancedPAdicInteger p

FullyPatternMatchable BalancedPAdicInteger p

GcdDomain

InnerEvalable(BalancedPAdicInteger p, BalancedPAdicInteger p) if BalancedPAdicInteger p has Evalable BalancedPAdicInteger p

InnerEvalable(Symbol, BalancedPAdicInteger p) if BalancedPAdicInteger p has InnerEvalable(Symbol, BalancedPAdicInteger p)

IntegralDomain

LeftModule %

LeftModule BalancedPAdicInteger p

LeftModule Fraction Integer

LeftOreRing

LinearlyExplicitOver BalancedPAdicInteger p

LinearlyExplicitOver Integer if BalancedPAdicInteger p has LinearlyExplicitOver Integer

Magma

MagmaWithUnit

Module %

Module BalancedPAdicInteger p

Module Fraction Integer

Monoid

NonAssociativeAlgebra %

NonAssociativeAlgebra BalancedPAdicInteger p

NonAssociativeAlgebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedAbelianMonoid if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedAbelianSemiGroup if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedCancellationAbelianMonoid if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedIntegralDomain if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedRing if BalancedPAdicInteger p has OrderedIntegralDomain

OrderedSet if BalancedPAdicInteger p has OrderedSet

PartialDifferentialRing Symbol if BalancedPAdicInteger p has PartialDifferentialRing Symbol

PartialOrder if BalancedPAdicInteger p has OrderedSet

Patternable BalancedPAdicInteger p

PatternMatchable Float if BalancedPAdicInteger p has PatternMatchable Float

PatternMatchable Integer if BalancedPAdicInteger p has PatternMatchable Integer

PolynomialFactorizationExplicit if BalancedPAdicInteger p has PolynomialFactorizationExplicit

PrincipalIdealDomain

QuotientFieldCategory BalancedPAdicInteger p

RealConstant if BalancedPAdicInteger p has RealConstant

RetractableTo BalancedPAdicInteger p

RetractableTo Fraction Integer if BalancedPAdicInteger p has RetractableTo Integer

RetractableTo Integer if BalancedPAdicInteger p has RetractableTo Integer

RetractableTo Symbol if BalancedPAdicInteger p has RetractableTo Symbol

RightModule %

RightModule BalancedPAdicInteger p

RightModule Fraction Integer

RightModule Integer if BalancedPAdicInteger p has LinearlyExplicitOver Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if BalancedPAdicInteger p has StepThrough

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown