Documentation

Introduction
Differential Rings and Fields
- Creation
- Creation of Differential Ring Elements
Structure Operations on Differential Rings
- Category and Parent
  - Category(R) : RngDiff -> RngDiff
  - Parent(R) : RngDiff -> PowStr
- Related Structures
- Derivation and Differential
- Numerical Invariants
  - Ngens(R) : RngDiff -> RngIntElt
- Predicates and Booleans
- Precision
Element Operations on Differential Ring Elements
- Category and Parent
  - Category(s) : RngDiffElt -> RngDiffElt
  - Parent(s) : RngDiffElt -> RngDiff
- Arithmetic
- Predicates and Booleans
- Coefficients and Terms
- Conjugates, Norm and Trace
  - MinimalPolynomial(s) : RngDiffElt -> RngUPolElt
  - Example RngDiff_Example-minimal-polynomial-differential-rings (H121E16)
- Derivatives and Differentials
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
- 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
- Creation of Differential Operators
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
- Predicates and Booleans
- 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
- Predicates and Booleans
- Coefficients and Terms
- Order and Degree
- Related Differential Operators
- 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
- Greatest Common Right and Left Divisors
- Least Common Left Multiples
Related Matrices
- CompanionMatrix(L) : RngDiffOpElt -> AlgMatElt
- Example RngDiff_example-companion-matrix (H121E55)
Singular Places and Indicial Polynomials
- Singular Places
- 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
- Right Hand Factors of Operators
Bibliography

V2.29, 21 October 2025