Power, Laurent, and Puiseux Series
- Introduction
- Creation Functions
- Creation of Structures
PowerSeriesRing(R): Rng → RngSerPowPowerSeriesRing(R, p): Rng, RngIntElt → RngSerPowLaurentSeriesRing(R): Rng → RngSerLaurLaurentSeriesRing(R, p): Rng, RngIntElt → RngSerLaurPuiseuxSeriesRing(R): Rng → RngSerPuisPuiseuxSeriesRing(R, p): Rng, RngIntElt → RngSerPuis- Example: Global and Non-Global Series Rings
- Special Options
AssertAttribute(S, "DefaultPrecision", n): RngSer, MonStgElt, RngIntEltHasAttribute(S, "DefaultPrecision"): RngSer, MonStgElt → BoolElt, RngIntEltAssignNames(~S, ["x"]): RngSer, [ MonStgElt ]Name(S, 1): RngSer, RngIntElt → RngSerEltS . 1: RngSer, RngIntElt → RngSerEltName(S, 1): RngSer, RngIntElt → RngSerEltS . 1: RngSer, RngIntElt → RngSerElt
- Creation of Elements
R . 1: RngSer, RngInt → RngSerEltUniformizingElement(R): RngSer → RngSerEltelt< R | v, [ a₁, ..., a_d], p >: RngIntElt, SeqEnum, RngIntElt → RngSerEltR ! s: RngSer, SeqEnum → RngSerEltBigO(f): RngSerElt → RngIntEltO(f): RngSerElt → RngIntEltOne(Q): RngSer → RngSerEltOne(Q): RngSer → RngSerEltIdentity(Q): RngSer → RngSerEltIdentity(Q): RngSer → RngSerEltZero(Q): RngSer → RngSerEltZero(Q): RngSer → RngSerEltRepresentative(Q): RngSer → RngSerEltRepresentative(Q): RngSer → RngSerElt
- Creation of Structures
- Structure Operations
- Related Structures
Parent(R): RngSer → PowCategory(R): RngSer → CatBaseRing(R): RngSer → RngCoefficientRing(R): RngSer → RngIntegerRing(R): RngSer → RngSerPowIntegers(R): RngSer → RngSerPowRingOfIntegers(R): RngSer → RngSerPowFieldOfFractions(R): RngSer → RngSerLaurChangePrecision(R, r): RngSer, Any → RngSerChangePrecision(~R, r): RngSer, Any → RngSerChangeRing(R, C): RngSer, Rng → RngSer, MapResidueClassField(R): RngSer → Rng, Map
- Invariants
- Ring Predicates and Booleans
IsCommutative(Q): RngSer → BoolEltIsUnitary(Q): RngSer → BoolEltIsFinite(Q): RngSer → BoolEltIsOrdered(Q): RngSer → BoolEltIsField(Q): RngSer → BoolEltIsEuclideanDomain(Q): RngSer → BoolEltIsPID(Q): RngSer → BoolEltIsUFD(Q): RngSer → BoolEltIsDivisionRing(Q): RngSer → BoolEltIsEuclideanRing(Q): RngSer → BoolEltIsPrincipalIdealRing(Q): RngSer → BoolEltIsDomain(Q): RngSer → BoolEltR eq S: RngSer, RngSer → BoolEltR ne S: RngSer, RngSer → BoolElt
- Related Structures
- Basic Element Operations
- Parent and Category
- Arithmetic Operators
+ b: RngSerElt → RngSerElt- b: RngSerElt → RngSerElta + b: RngSerElt, RngSerElt → RngSerElta - b: RngSerElt, RngSerElt → RngSerElta * b: RngSerElt, RngSerElt → RngSerElta ^ k: RngSerElt, RngIntElt → RngSerElta div b: RngSerPowElt, RngSerPowElt → RngSerPowElta / b: RngSerElt, RngSerElt → RngSerElt
- Equality and Membership
- Predicates on Ring Elements
IsZero(a): RngSerElt → BoolEltIsOne(a): RngSerElt → BoolEltIsMinusOne(a): RngSerElt → BoolEltIsNilpotent(x): RngSerElt → BoolEltIsIdempotent(x): RngSerElt → BoolEltIsUnit(a): RngSerElt → BoolEltIsZeroDivisor(x): RngSerElt → BoolEltIsRegular(x): RngSerElt → BoolEltIsIrreducible(x): RngSerElt → BoolEltIsPrime(x): RngSerElt → BoolEltIsWeaklyZero(f): RngSerElt → BoolEltIsWeaklyEqual(f, g): RngSerElt, RngSerElt → BoolEltIsIdentical(f, g): RngSerElt, RngSerElt → BoolElt
- Precision
- Coefficients and Degree
Coefficients(f): RngSerElt → [ RngElt ], RngIntElt, RngIntEltElementToSequence(f): RngSerElt → [ RngElt ], RngIntElt, RngIntEltEltseq(f): RngSerElt → [ RngElt ], RngIntElt, RngIntEltCoefficient(f, i): RngSerElt, RngElt → RngEltLeadingCoefficient(f): RngSerElt → RngEltLeadingTerm(f): RngSerElt → RngEltTruncate(f): RngSerElt → RngSerEltExponentDenominator(f): RngMSerElt → RngEltDegree(f): RngSerElt → RngIntEltValuation(f): RngSerElt → RngIntEltExponentDenominator(f): RngSerElt → RngIntElt
- Evaluation and Derivative
- Square Root
- Composition and Reversion
- Transcendental Functions
- The Hypergeometric Series
- Polynomials over Series Rings
HenselLift(f, L): RngUPolElt[RngSer], SeqEnum[RngUPolElt] → [RngUPolElt]HenselLift(f, L): RngUPolElt[RngSerExt], SeqEnum[RngUPolElt] → [RngUPolElt]Factorization(f): RngUPolElt[RngSerPow[FldFin]] → [ < RngUPolElt[RngSerPow], RngIntElt > ], RngSerPowEltFactorization(f): RngUPolElt[RngSerLaur[FldFin]] → [ < RngUPolElt[RngSerLaur], RngIntElt > ], RngSerLaurEltFactorization(f): RngUPolElt[RngSerExt] → [ < RngUPolElt[RngSerExt], RngIntElt > ], RngSerExtElt- Example: Factorization over Series Rings
- Extensions of Series Rings
- Constructions of Extensions
UnramifiedExtension(R, f): RngSerPow[FldFin], RngUPolElt → RngSerExtUnramifiedExtension(R, f): RngSerLaur[FldFin], RngUPolElt → RngSerExtUnramifiedExtension(R, f): RngSerExt, RngUPolElt → RngSerExtTotallyRamifiedExtension(R, f): RngSerPow[FldFin], RngUPolElt → RngSerExtTotallyRamifiedExtension(R, f): RngSerLaur[FldFin], RngUPolElt → RngSerExtTotallyRamifiedExtension(R, f): RngSerExt, RngUPolElt → RngSerExtChangePrecision(E, r): RngSerExt, RngIntElt → RngSerExtChangePrecision(~E, r): RngSerExt, RngIntEltFieldOfFractions(E): RngSerExt → RngSerExt- Example: Two-Step Extension of a Series Ring
- Operations on Extensions
Precision(E): RngSerExt → RngIntEltGetPrecision(E): RngSerExt → RngIntEltCoefficientRing(E): RngSerExt → RngBaseRing(E): RngSerExt → RngDefiningPolynomial(E): RngSerExt → RngUPolEltInertiaDegree(E): RngSerExt → RngIntEltRamificationIndex(E): RngSerExt → RngIntEltRamificationDegree(E): RngSerExt → RngIntEltResidueClassField(E): RngSerExt → FldFinUniformizingElement(E): RngSerExt → RngSerExtEltIntegerRing(E): RngSerExt → RngSerExtIntegers(E): RngSerExt → RngSerExtRingOfIntegers(E): RngSerExt → RngSerExtE1 eq E2: RngSerExt, RngSerExt → BoolEltE . i: RngSerExt, RngIntElt → RngSerExtEltAssignNames(~E, S): RngSerExt, [ MonStgElt ]- Example: Operations on Series Ring Extensions
- Elements of Extensions
x * y: RngSerExtElt, RngSerExtElt → RngSerExtEltx + y: RngSerExtElt, RngSerExtElt → RngSerExtEltx - y: RngSerExtElt, RngSerExtElt → RngSerExtElt- x: RngSerExtElt → RngSerExtEltx ^ n: RngSerExtElt, RngIntElt → RngSerExtEltx div y: RngSerExtElt, RngSerExtElt → RngSerExtEltx / y: RngSerExtElt, RngSerExtElt → RngSerExtEltx eq y: RngSerExtElt, RngSerExtElt → BoolEltIsZero(e): RngSerExtElt → BoolEltIsOne(e): RngSerExtElt → BoolEltIsMinusOne(e): RngSerExtElt → BoolEltIsUnit(e): RngSerExtElt → BoolEltValuation(e): RngSerExtElt → RngIntEltRelativePrecision(e): RngSerExtElt → RngIntEltAbsolutePrecision(e): RngSerExtElt → RngIntEltCoefficients(e): RngSerExtElt → [ RngElt ]Eltseq(e): RngSerExtElt → [ RngElt ]ElementToSequence(e): RngSerExtElt → [ RngElt ]- Example: Arithmetic in Series Ring Extensions
- Optimized Representation
- Constructions of Extensions
- Bibliography