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DIFFERENTIAL RINGS

 
Acknowledgements
 
Introduction
 
Differential Rings and Fields
      Creation
      Creation of Differential Ring Elements
 
Structure Operations on Differential Rings
      Category and Parent
      Related Structures
      Derivation and Differential
      Numerical Invariants
      Predicates and Booleans
      Precision
 
Element Operations on Differential Ring Elements
      Category and Parent
      Arithmetic
      Predicates and Booleans
      Coefficients and Terms
      Conjugates, Norm and Trace
      Derivatives and Differentials
 
Changing Related Structures
 
Ring and Field Extensions
 
Ideals and Quotient Rings
      Defining Ideals and Quotient Rings
      Boolean Operations on Ideals
 
Wronskian Matrix
 
Differential Operator Rings
      Creation
      Creation of Differential Operators
 
Structure Operations on Differential Operator Rings
      Category and Parent
      Related Structures
      Derivation and Differential
      Predicates and Booleans
      Precision
 
Element Operations on Differential Operators
      Category and Parent
      Arithmetic
      Predicates and Booleans
      Coefficients and Terms
      Order and Degree
      Related Differential Operators
      Application of Operators
 
Related Maps
 
Changing Related Structures
 
Euclidean Algorithms, GCDs and LCMs
      Euclidean Right and Left Division
      Greatest Common Right and Left Divisors
      Least Common Left Multiples
 
Related Matrices
 
Singular Places and Indicial Polynomials
      Singular Places
      Indicial Polynomials
 
Rational Solutions
 
Newton Polygons
 
Symmetric Powers
 
Differential Operators of Algebraic Functions
 
Factorisation of Operators over Differential Laurent Series Rings
      Slope Valuation of an Operator
      Coprime Index 1 and LCLM Factorisation
      Right Hand Factors of Operators
 
Bibliography







DETAILS

 
Introduction

 
Differential Rings and Fields

      Creation
            DifferentialRing(P, f, C) : Rng, Map, Rng -> RngDiff
            Example RngDiff_diff-ring-create (H111E1)
            RationalDifferentialField(C) : Fld -> RngDiff
            Example RngDiff_rat-diff-field-create (H111E2)
            DifferentialLaurentSeriesRing(C) : Fld -> RngDiff
            Example RngDiff_diff-laur-ser-ring-create (H111E3)
            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 (H111E4)

 
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 (H111E5)
            Example RngDiff_diff-laur-ser-related-structures (H111E6)

      Derivation and Differential
            Derivation(R) : RngDiff -> Map
            Differential(F) : RngDiff -> DiffFunElt
            Example RngDiff_diff-ring-derivation-differential (H111E7)

      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 (H111E8)
            HasProjectiveDerivation(F) : RngDiff -> BoolElt
            HasZeroDerivation(F) : RngDiff -> BoolElt
            Example RngDiff_diff-rings-booleans-derivation (H111E9)

      Precision
            RelativePrecision(F) : RngDiff -> RngElt
            RelativePrecisionOfDerivation(F) : RngDiff -> RngElt
            Example RngDiff_diff-rings-relative-precision (H111E10)
            Example RngDiff_Example-differential-ring-relative-precision-derivation (H111E11)
            ChangePrecision(F, p) : RngDiff, RngElt -> RngDiff, Map
            Example RngDiff_diff-rings-change-precision (H111E12)

 
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 (H111E13)

      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 (H111E14)
            Example RngDiff_Example-coefficients-terms-differential-series (H111E15)

      Conjugates, Norm and Trace
            MinimalPolynomial(s) : RngDiffElt -> RngUPolElt
            Example RngDiff_Example-minimal-polynomial-differential-rings (H111E16)

      Derivatives and Differentials
            Derivative(s) : RngDiffElt -> RngDiffElt
            Differential(s) : RngDiffElt -> RngDiffElt
            Example RngDiff_derivative-differential-diff-ring-elements (H111E17)

 
Changing Related Structures
      ChangeDerivation(R, f) : RngDiff, RngElt -> RngDiff, Map
      Example RngDiff_diff-ring-change-derivation (H111E18)
      ChangeDifferential(F, df) : RngDiff, DiffFunElt -> RngDiff, Map
      Example RngDiff_diff-ring-change-differential (H111E19)
      ConstantFieldExtension(F, C) : RngDiff, Fld -> RngDiff, Map
      Example RngDiff_diff-ring-constant-field-extension (H111E20)
      Example RngDiff_diff-ring-constant-field-extension-series (H111E21)
      Completion(F, p) : RngDiff, PlcFunElt -> RngDiff, Map
      Example RngDiff_diff-ring-completion-create (H111E22)
      Example RngDiff_diff-ring-completion-elliptic (H111E23)

 
Ring and Field Extensions
      DifferentialRingExtension(L) : RngDiffOpElt -> RngDiff
      DifferentialFieldExtension(L) : RngDiffOpElt, -> RngDiff
      Example RngDiff_diff-ring-ext-operator (H111E24)
      Example RngDiff_diff-field-ext-operator (H111E25)
      ext< F | f > : RngDiff, RngUPolElt -> RngDiff
      Example RngDiff_diff-field-ext-ext (H111E26)
      ExponentialFieldExtension(F, f) : RngDiff, RngDiffElt -> RngDiff
      LogarithmicFieldExtension(F, f) : RngDiff, RngDiffElt -> RngDiff
      Example RngDiff_diff-field-ext-exp-log (H111E27)
      PurelyRamifiedExtension(f) : RngUPolElt[RngDiff] -> RngDiff, Map
      Example RngDiff_diff-field-purely-ramified-ext-ADF (H111E28)
      Example RngDiff_diff-field-purely-ramified-ext-DLSR-1 (H111E29)
      Example RngDiff_diff-field-purely-ramified-ext-DLSR-2 (H111E30)

 
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 (H111E31)

      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 (H111E32)

 
Differential Operator Rings

      Creation
            DifferentialOperatorRing(F) : RngDiff -> RngDiffOp
            Example RngDiff_create-diff-op-ring (H111E33)
            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 (H111E34)

 
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 (H111E35)

      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 (H111E36)
            Example RngDiff_diff-op-ring_booleans-LSR (H111E37)

      Precision
            RelativePrecisionOfDerivation(R) : RngDiffOp -> RngElt
            Example RngDiff_diff-op-rings-relative-precision (H111E38)

 
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 (H111E39)

      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 (H111E40)

      Order and Degree
            Order(L) : RngDiffOpElt -> RngIntElt
            WeakOrder(L) : RngDiffOpElt -> RngIntElt
            Example RngDiff_example-booleans-degrees-diff-ops (H111E41)

      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 (H111E42)
            Example RngDiff_related-diff-op-truncate-coefficients (H111E43)

      Application of Operators
            Apply(L, f) : RngDiffOpElt, RngElt -> RngElt
            Example RngDiff_example-apply (H111E44)

 
Related Maps
      TranslationMap(R, e) : RngDiffOp, RngElt -> Map
      LiftMap(m, R) : Map, RngDiffOp -> Map
      Example RngDiff_example-maps-between-diff-op-rings (H111E45)

 
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 (H111E46)
      ConstantFieldExtension(R, C) : RngDiffOp,Fld -> RngDiffOp, Map
      PurelyRamifiedExtension(R,f) : RngDiffOp,RngUPolElt -> RngDiffOp, Map
      Example RngDiff_diff-op-ring-purely-ramified-extension-DLSR (H111E47)
      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 (H111E48)
      Example RngDiff_example-diff-op-ring-completion (H111E49)
      Example RngDiff_example-diff-op-ring-localization (H111E50)

 
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 (H111E51)

      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 (H111E52)

      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 (H111E53)
            Example RngDiff_example-LCLM-conjugates (H111E54)

 
Related Matrices
      CompanionMatrix(L) : RngDiffOpElt -> AlgMatElt
      Example RngDiff_example-companion-matrix (H111E55)

 
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 (H111E56)
            Example RngDiff_example-regular-singular-DLSR (H111E57)

      Indicial Polynomials
            IndicialPolynomial(L, p) : RngDiffOpElt, PlcFunElt -> RngElt
            Example RngDiff_example-indicial-pol (H111E58)

 
Rational Solutions
      RationalSolutions(L) : RngDiffOpElt, -> SeqEnum
      HasRationalSolutions(L, g) : RngDiffOpElt, RngElt -> BoolElt, RngElt, SeqEnum
      Example RngDiff_example-rational-sols (H111E59)

 
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 (H111E60)
      Example RngDiff_example-newton-pgns-2 (H111E61)
      Example RngDiff_example-newton-pgns-3 (H111E62)

 
Symmetric Powers
      SymmetricPower(L, m) : RngDiffOpElt, RngIntElt -> RngDiffOpElt
      Example RngDiff_example-symmetric-power (H111E63)

 
Differential Operators of Algebraic Functions
      DifferentialOperator(f) : RngUPolElt -> RngDiffOpElt
      Example RngDiff_example-diff-op-algebraic-function (H111E64)

 
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 (H111E65)

      Coprime Index 1 and LCLM Factorisation
            Factorisation(L) : RngDiffOpElt -> SeqEnum, SeqEnum
            Example RngDiff_example-diff-op-factorisation-LCLM-1 (H111E66)
            Example RngDiff_example-diff-op-factorisation-LCLM-2 (H111E67)
            Example RngDiff_example-diff-op-factorisation-LCLM-3 (H111E68)
            Example RngDiff_example-diff-op-factorisation-LCLM-4 (H111E69)
            Example RngDiff_example-diff-op-factorisation-LCLM-5 (H111E70)
            Example RngDiff_example-diff-op-factorisation-LCLM-6 (H111E71)

      Right Hand Factors of Operators
            RightHandFactors(L) : RngDiffOpElt -> SeqEnum, SeqEnum[[BoolElt]]
            Example RngDiff_example-diff-op-righthandfactors-1 (H111E72)
            Example RngDiff_example-diff-op-righthandfactors-2 (H111E73)
            Example RngDiff_example-diff-op-righthandfactors-3 (H111E74)
            Example RngDiff_example-diff-op-righthandfactors-4 (H111E75)
            Example RngDiff_example-diff-op-righthandfactors-5 (H111E76)
            Example RngDiff_example-diff-op-righthandfactors-5 (H111E77)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013