Introduction

 
Differential Rings and Fields

      Creation
            DifferentialRing(P, f, C) : Rng, Map, Rng -> RngDiff
            Example RngDiff_diff-ring-create (H121E1)
            RationalDifferentialField(C) : Fld -> RngDiff
            Example RngDiff_rat-diff-field-create (H121E2)
            DifferentialLaurentSeriesRing(C) : Fld -> RngDiff
            Example RngDiff_diff-laur-ser-ring-create (H121E3)
            RingOfFractions(R) : RngDiff -> RngDiff, Map
            FieldOfFractions(R) : RngDiff -> RngDiff, Map
            AssignNames(~R, S) : RngDiff, [MonStgElt] ->

      Creation of Differential Ring Elements
            Name(R, i) : RngDiff, RngIntElt -> RngDiffElt
            R ! s : RngDiff, RngElt -> RngDiffElt
            Zero(R) : RngDiff -> RngDiffElt
            One(R) : RngDiff -> RngDiffElt
            SeparatingElement(F) : RngDiff -> RngDiffElt
            Example RngDiff_diff-ring-element-creation (H121E4)

 
Structure Operations on Differential Rings

      Category and Parent
            Category(R) : RngDiff -> RngDiff
            Parent(R) : RngDiff -> PowStr

      Related Structures
            UnderlyingRing(R) : RngDiff -> Rng
            UnderlyingField(R) : RngDiff -> Rng
            BaseRing(R) : RngDiff -> Rng
            BaseField(R) : RngDiff -> Rng
            ConstantRing(R) : RngDiff -> Rng
            ConstantField(R) : RngDiff -> Rng
            ExactConstantField(F) : RngDiff -> RngDiff, Map
            Generators(R) : RngDiff -> SeqEnum
            Example RngDiff_diff-ring-related-structures (H121E5)
            Example RngDiff_diff-laur-ser-related-structures (H121E6)

      Derivation and Differential
            Derivation(R) : RngDiff -> Map
            Differential(F) : RngDiff -> DiffFunElt
            Example RngDiff_diff-ring-derivation-differential (H121E7)

      Numerical Invariants
            Ngens(R) : RngDiff -> RngIntElt

      Predicates and Booleans
            R eq F : RngDiff, RngDiff -> BoolElt
            IsIdentical(R, F) : RngDiff, RngDiff -> BoolElt
            IsDomain(R) : RngDiff -> BoolElt
            IsField(R) : RngDiff -> BoolElt
            IsDifferentialField(R) : Rng -> BoolElt
            IsAlgebraicDifferentialField(R) : Rng -> BoolElt
            IsDifferentialSeriesRing(R) : Rng -> BoolElt
            IsDifferentialLaurentSeriesRing(R) : Rng -> BoolElt
            Example RngDiff_diff-rings-booleans (H121E8)
            HasProjectiveDerivation(F) : RngDiff -> BoolElt
            HasZeroDerivation(F) : RngDiff -> BoolElt
            Example RngDiff_diff-rings-booleans-derivation (H121E9)

      Precision
            RelativePrecision(F) : RngDiff -> RngElt
            RelativePrecisionOfDerivation(F) : RngDiff -> RngElt
            Example RngDiff_diff-rings-relative-precision (H121E10)
            Example RngDiff_Example-differential-ring-relative-precision-derivation (H121E11)
            ChangePrecision(F, p) : RngDiff, RngElt -> RngDiff, Map
            Example RngDiff_diff-rings-change-precision (H121E12)

 
Element Operations on Differential Ring Elements

      Category and Parent
            Category(s) : RngDiffElt -> RngDiffElt
            Parent(s) : RngDiffElt -> RngDiff

      Arithmetic
            s + t : RngDiffElt, RngDiffElt -> RngDiffElt
            - s : RngDiffElt -> RngDiffElt
            s - t : RngDiffElt, RngDiffElt -> RngDiffElt
            s * t : RngDiffElt, RngDiffElt -> RngDiffElt
            s ^ n : RngDiffElt, RngIntElt -> RngDiffElt
            s div t : RngDiffElt, RngDiffElt -> RngDiffElt
            s / t : RngDiffElt, RngDiffElt -> RngDiffElt

      Predicates and Booleans
            s eq t : RngDiffElt, RngDiffElt -> BoolElt
            IsZero(s) : RngDiffElt -> BoolElt
            IsOne(s) : RngDiffElt -> BoolElt
            IsWeaklyEqual(s, t) : RngDiffElt, RngDiffElt -> BoolElt
            IsWeaklyZero(s) : RngDiffElt -> BoolElt
            IsOrderTerm(s) : RngDiffElt -> BoolElt
            Example RngDiff_diff-ring-elts-booleans (H121E13)

      Coefficients and Terms
            O(s) : RngDiffElt -> RngDiffElt
            Truncate(s) : RngDiffElt -> RngDiffElt
            Eltseq(s) : RngDiffElt -> SeqEnum
            Exponents(s) : RngDiffElt -> SeqEnum
            Example RngDiff_Example-coefficients-terms-algebraic-differential-elements (H121E14)
            Example RngDiff_Example-coefficients-terms-differential-series (H121E15)

      Conjugates, Norm and Trace
            MinimalPolynomial(s) : RngDiffElt -> RngUPolElt
            Example RngDiff_Example-minimal-polynomial-differential-rings (H121E16)

      Derivatives and Differentials
            Derivative(s) : RngDiffElt -> RngDiffElt
            Differential(s) : RngDiffElt -> RngDiffElt
            Example RngDiff_derivative-differential-diff-ring-elements (H121E17)

 
Changing Related Structures
      ChangeDerivation(R, f) : RngDiff, RngElt -> RngDiff, Map
      Example RngDiff_diff-ring-change-derivation (H121E18)
      ChangeDifferential(F, df) : RngDiff, DiffFunElt -> RngDiff, Map
      Example RngDiff_diff-ring-change-differential (H121E19)
      ConstantFieldExtension(F, C) : RngDiff, Fld -> RngDiff, Map
      Example RngDiff_diff-ring-constant-field-extension (H121E20)
      Example RngDiff_diff-ring-constant-field-extension-series (H121E21)
      Completion(F, p) : RngDiff, PlcFunElt -> RngDiff, Map
      Example RngDiff_diff-ring-completion-create (H121E22)
      Example RngDiff_diff-ring-completion-elliptic (H121E23)

 
Ring and Field Extensions
      DifferentialRingExtension(L) : RngDiffOpElt -> RngDiff
      DifferentialFieldExtension(L) : RngDiffOpElt -> RngDiff
      Example RngDiff_diff-ring-ext-operator (H121E24)
      Example RngDiff_diff-field-ext-operator (H121E25)
      ext< F | f > : RngDiff, RngUPolElt -> RngDiff
      Example RngDiff_diff-field-ext-ext (H121E26)
      ExponentialFieldExtension(F, f) : RngDiff, RngDiffElt -> RngDiff
      LogarithmicFieldExtension(F, f) : RngDiff, RngDiffElt -> RngDiff
      Example RngDiff_diff-field-ext-exp-log (H121E27)
      PurelyRamifiedExtension(f) : RngUPolElt[RngDiff] -> RngDiff, Map
      Example RngDiff_diff-field-purely-ramified-ext-ADF (H121E28)
      Example RngDiff_diff-field-purely-ramified-ext-DLSR-1 (H121E29)
      Example RngDiff_diff-field-purely-ramified-ext-DLSR-2 (H121E30)

 
Ideals and Quotient Rings

      Defining Ideals and Quotient Rings
            DifferentialIdeal(L) : [RngDiffElt] -> RngMPol
            QuotientRing(R, I) : RngDiff, RngMPol -> RngDiff, Map
            Example RngDiff_diff-ideal-quotient (H121E31)

      Boolean Operations on Ideals
            IsDifferentialIdeal(R, I) : RngDiff, RngMPol -> BoolElt

 
Wronskian Matrix
      WronskianMatrix(L) : [RngDiffElt] -> AlgMatElt
      WronskianDeterminant(L) : [RngDiffElt] -> RngDiffElt, AlgMatElt
      Example RngDiff_wronskian-mat-det (H121E32)

 
Differential Operator Rings

      Creation
            DifferentialOperatorRing(F) : RngDiff -> RngDiffOp
            Example RngDiff_create-diff-op-ring (H121E33)
            AssignNames(~R, S) : RngDiffOp, [MonStgElt] ->

      Creation of Differential Operators
            Name(R, i) : RngDiffOp, RngIntElt -> RngDiffElt
            R ! s : RngDiffOp, RngElt -> RngDiffOpElt
            Zero(R) : RngDiffOp -> RngDiffOpElt
            One(R) : RngDiffOp -> RngDiffOpElt
            Example RngDiff_example-diff-op-coercion (H121E34)

 
Structure Operations on Differential Operator Rings

      Category and Parent
            Category(R) : RngDiffOp -> RngDiffOp
            Parent(R) : RngDiffOp -> PowStr

      Related Structures
            BaseRing(R) : RngDiffOp -> Rng
            ConstantRing(R) : RngDiffOp -> Rng

      Derivation and Differential
            Derivation(R) : RngDiffOp -> Map
            Differential(R) : RngDiffOp -> DiffFunElt
            Example RngDiff_diff-op-ring_related_structures (H121E35)

      Predicates and Booleans
            R eq F : RngDiffOp, RngDiffOp -> BoolElt
            IsIdentical(R, F) : RngDiffOp, RngDiffOp -> BoolElt
            IsDifferentialOperatorRing(R) : . -> BoolElt
            HasProjectiveDerivation(R) : RngDiffOp -> BoolElt
            HasZeroDerivation(R) : RngDiffOp -> BoolElt
            Example RngDiff_diff-op-ring_booleans (H121E36)
            Example RngDiff_diff-op-ring_booleans-LSR (H121E37)

      Precision
            RelativePrecisionOfDerivation(R) : RngDiffOp -> RngElt
            Example RngDiff_diff-op-rings-relative-precision (H121E38)

 
Element Operations on Differential Operators

      Category and Parent
            Category(L) : RngDiffOpElt -> RngDiffOpElt
            Parent(L) : RngDiffOpElt -> RngDiffOp

      Arithmetic
            s + t : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt
            - s : RngDiffOpElt -> RngDiffOpElt
            s - t : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt
            s * t : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt
            s ^ n : RngDiffOpElt, RngIntElt -> RngDiffElt
            Example RngDiff_diff-op-arithmetic (H121E39)

      Predicates and Booleans
            s eq t : RngDiffOpElt, RngDiffOpElt -> BoolElt
            IsZero(L) : RngDiffOpElt -> BoolElt
            IsOne(L) : RngDiffOpElt -> BoolElt
            IsMonic(L) : RngDiffOpElt -> BoolElt
            IsWeaklyEqual(L, P) : RngDiffOpElt, RngDiffOpElt -> BoolElt
            IsWeaklyZero(L) : RngDiffOpElt -> BoolElt
            IsWeaklyMonic(L) : RngDiffOpElt -> BoolElt

      Coefficients and Terms
            Eltseq(L) : RngDiffOpElt -> SeqEnum
            Coefficient(L, i) : RngDiffOpElt, RngIntElt -> RngElt
            LeadingCoefficient(L) : RngDiffOpElt -> RngElt
            LeadingTerm(L) : RngDiffOpElt -> RngDiffOpElt
            Terms(L) : RngDiffOpElt -> SeqEnum
            Example RngDiff_diff-op-coeff-terms (H121E40)

      Order and Degree
            Order(L) : RngDiffOpElt -> RngIntElt
            WeakOrder(L) : RngDiffOpElt -> RngIntElt
            Example RngDiff_example-booleans-degrees-diff-ops (H121E41)

      Related Differential Operators
            MonicDifferentialOperator(L) : RngDiffOpElt -> RngDiffOpElt
            Adjoint(L) : RngDiffOpElt -> RngDiffOpElt
            Translation(L, e) : RngDiffOpElt, RngElt -> RngDiffOpElt, Map
            TruncateCoefficients(L) : RngDiffOpElt -> RngDiffOpElt
            Example RngDiff_related-diff-op (H121E42)
            Example RngDiff_related-diff-op-truncate-coefficients (H121E43)

      Application of Operators
            Apply(L, f) : RngDiffOpElt, RngElt -> RngElt
            Example RngDiff_example-apply (H121E44)

 
Related Maps
      TranslationMap(R, e) : RngDiffOp, RngElt -> Map
      LiftMap(m, R) : Map, RngDiffOp -> Map
      Example RngDiff_example-maps-between-diff-op-rings (H121E45)

 
Changing Related Structures
      ChangeDerivation(R, f) : RngDiffOp, RngElt -> RngDiffOp, Map
      ChangeDifferential(R, df) : RngDiffOp, DiffFunElt -> RngDiffOp, Map
      Example RngDiff_example-diff-op-ring-changing-differential (H121E46)
      ConstantFieldExtension(R, C) : RngDiffOp,Fld -> RngDiffOp, Map
      PurelyRamifiedExtension(R,f) : RngDiffOp,RngUPolElt -> RngDiffOp, Map
      Example RngDiff_diff-op-ring-purely-ramified-extension-DLSR (H121E47)
      Completion(R, p) : RngDiffOp, PlcFunElt -> RngDiffOp, Map
      Localization(R, p) : RngDiffOp, PlcFunElt -> RngDiffOp, Map, PlcFunElt
      Localization(L, p) : RngDiffOpElt, PlcFunElt -> RngDiffOpElt, Map, PlcFunElt
      Localization(R) : RngDiffOp -> RngDiffOp, Map
      Localization(L) : RngDiffOpElt -> RngDiffOpElt, Map
      Example RngDiff_example-diff-op-ring-changing-attributes-DLSR (H121E48)
      Example RngDiff_example-diff-op-ring-completion (H121E49)
      Example RngDiff_example-diff-op-ring-localization (H121E50)

 
Euclidean Algorithms, GCDs and LCMs

      Euclidean Right and Left Division
            EuclideanRightDivision(N, D) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt,RngDiffOpElt
            EuclideanLeftDivision(D, N) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt,RngDiffOpElt
            Example RngDiff_example-eucl-alg (H121E51)

      Greatest Common Right and Left Divisors
            GreatestCommonRightDivisor(A, B) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt
            ExtendedGreatestCommonRightDivisor(A, B) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt, RngDiffOpElt, RngDiffOpElt
            GreatestCommonLeftDivisor(A, B) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt
            ExtendedGreatestCommonLeftDivisor(A, B) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt, RngDiffOpElt, RngDiffOpElt
            Example RngDiff_example-GCRD-GCLD (H121E52)

      Least Common Left Multiples
            LeastCommonLeftMultiple(L) : RngDiffOpElt -> RngDiffOpElt
            LeastCommonLeftMultiple(A, B) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt
            ExtendedLeastCommonLeftMultiple(A, B) : RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt, RngDiffOpElt, RngDiffOpElt
            ExtendedLeastCommonLeftMultiple(S) : [RngDiffOpElt] -> RngDiffOpElt, SeqEnum
            Example RngDiff_example-LCLM (H121E53)
            Example RngDiff_example-LCLM-conjugates (H121E54)

 
Related Matrices
      CompanionMatrix(L) : RngDiffOpElt -> AlgMatElt
      Example RngDiff_example-companion-matrix (H121E55)

 
Singular Places and Indicial Polynomials

      Singular Places
            IsRegularPlace(L, p) : RngDiffOpElt, PlcFunElt -> BoolElt
            IsRegularSingularPlace(L, p) : RngDiffOpElt, PlcFunElt -> BoolElt
            IsIrregularSingularPlace(L, p) : RngDiffOpElt, PlcFunElt -> BoolElt
            SetsOfSingularPlaces(L) : RngDiffOpElt -> SetEnum, SetEnum
            IsFuchsianOperator(L) : RngDiffOpElt -> BoolElt, SetEnum
            IsRegularSingularOperator(L) : RngDiffOpElt -> BoolElt, SetEnum
            Example RngDiff_example-singularities (H121E56)
            Example RngDiff_example-regular-singular-DLSR (H121E57)

      Indicial Polynomials
            IndicialPolynomial(L, p) : RngDiffOpElt, PlcFunElt -> RngElt
            Example RngDiff_example-indicial-pol (H121E58)

 
Rational Solutions
      RationalSolutions(L) : RngDiffOpElt -> SeqEnum
      HasRationalSolutions(L, g) : RngDiffOpElt, RngElt -> BoolElt, RngElt, SeqEnum
      Example RngDiff_example-rational-sols (H121E59)

 
Newton Polygons
      NewtonPolygon(L) : RngDiffOpElt -> NwtnPgon, RingDiffOpElt
      NewtonPolygon(L, p) : RngDiffOpElt, PlcFunElt -> NwtnPgon, RingDiffOpElt
      NewtonPolynomial(F) : NwtnPgonFace -> RngUPolElt
      NewtonPolynomials(L) : RngDiffOpElt -> SeqEnum, SeqEnum
      Example RngDiff_example-newton-pgns-1 (H121E60)
      Example RngDiff_example-newton-pgns-2 (H121E61)
      Example RngDiff_example-newton-pgns-3 (H121E62)

 
Symmetric Powers
      SymmetricPower(L, m) : RngDiffOpElt, RngIntElt -> RngDiffOpElt
      Example RngDiff_example-symmetric-power (H121E63)

 
Differential Operators of Algebraic Functions
      DifferentialOperator(f) : RngUPolElt -> RngDiffOpElt
      Example RngDiff_example-diff-op-algebraic-function (H121E64)

 
Factorisation of Operators over Differential Laurent Series Rings

      Slope Valuation of an Operator
            SlopeValuation(L,s) : RngDiffOpElt, RngElt -> FldRatElt
            Example RngDiff_example-diff-op-slope-valuation (H121E65)

      Coprime Index 1 and LCLM Factorisation
            Factorisation(L) : RngDiffOpElt -> SeqEnum, SeqEnum
            Example RngDiff_example-diff-op-factorisation-LCLM-1 (H121E66)
            Example RngDiff_example-diff-op-factorisation-LCLM-2 (H121E67)
            Example RngDiff_example-diff-op-factorisation-LCLM-3 (H121E68)
            Example RngDiff_example-diff-op-factorisation-LCLM-4 (H121E69)
            Example RngDiff_example-diff-op-factorisation-LCLM-5 (H121E70)
            Example RngDiff_example-diff-op-factorisation-LCLM-6 (H121E71)

      Right Hand Factors of Operators
            RightHandFactors(L) : RngDiffOpElt -> SeqEnum, [BoolElt]
            Example RngDiff_example-diff-op-righthandfactors-1 (H121E72)
            Example RngDiff_example-diff-op-righthandfactors-2 (H121E73)
            Example RngDiff_example-diff-op-righthandfactors-3 (H121E74)
            Example RngDiff_example-diff-op-righthandfactors-4 (H121E75)
            Example RngDiff_example-diff-op-righthandfactors-5 (H121E76)
            Example RngDiff_example-diff-op-righthandfactors-5 (H121E77)

 
Bibliography

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