Differential Rings

• Introduction
• Differential Rings and Fields
  • Creation
    • DifferentialRing(P, f, C): Rng, Map, Rng → RngDiff
    • Example: Creating a General Differential Ring
    • RationalDifferentialField(C): Fld → RngDiff
    • Example: Creating a Rational Differential Field
    • DifferentialLaurentSeriesRing(C): Fld → RngDiff
    • Example: Creating a Differential Laurent Series Ring
    • 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 . i: RngDiff, RngIntElt → RngDiffElt
    • R ! s: RngDiff, RngElt → RngDiffElt
    • Zero(R): RngDiff → RngDiffElt
    • One(R): RngDiff → RngDiffElt
    • Identity(R): RngDiff → RngDiffElt
    • SeparatingElement(F): RngDiff → RngDiffElt
    • Example: Creating Differential Field Elements
• Structure Operations on Differential Rings
  • Category and Parent
    • Category(R): RngDiff → RngDiff
    • Type(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: Related Structures of a Differential Field
    • Example: Related Structures of a Laurent Series Ring
  • Derivation and Differential
    • Derivation(R): RngDiff → Map
    • Differential(F): RngDiff → DiffFunElt
    • Example: Derivation and Differential of a Field
  • 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: Predicates for Differential Rings
    • HasProjectiveDerivation(F): RngDiff → BoolElt
    • HasZeroDerivation(F): RngDiff → BoolElt
    • Example: Predicates on the Derivation
  • Precision
    • RelativePrecision(F): RngDiff → RngElt
    • RelativePrecisionOfDerivation(F): RngDiff → RngElt
    • Example: Relative Precision of a Series Ring
    • Example: Relative Precision of a Derivation
    • ChangePrecision(F, p): RngDiff, RngElt → RngDiff, Map
    • Example: Changing the Precision of a Series Ring
• Element Operations on Differential Ring Elements
  • Category and Parent
    • Category(s): RngDiffElt → RngDiffElt
    • Type(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
    • IsOrderTerm(s): RngSerElt → BoolElt
    • Example: Predicates for Differential Ring Elements
  • Coefficients and Terms
    • O(s): RngDiffElt → RngDiffElt
    • Truncate(s): RngDiffElt → RngDiffElt
    • Eltseq(s): RngDiffElt → SeqEnum
    • Exponents(s): RngDiffElt → SeqEnum
    • Exponents(s): RngSerElt → SeqEnum
    • Example: Coefficients of Algebraic Differential Elements
    • Example: Coefficients of Differential Series
  • Conjugates, Norm, and Trace
    • MinimalPolynomial(s): RngDiffElt → RngUPolElt
    • Example: Minimal Polynomial of an Algebraic Element
  • Derivatives and Differentials
    • Derivative(s): RngDiffElt → RngDiffElt
    • Differential(s): RngDiffElt → RngDiffElt
    • Example: Derivatives and Differentials of Elements
• Changing Related Structures
  • ChangeDerivation(R, f): RngDiff, RngElt → RngDiff, Map
  • Example: Changing the Derivation of a Differential Ring
  • ChangeDifferential(F, df): RngDiff, DiffFunElt → RngDiff, Map
  • Example: Changing the Differential of a Differential Ring
  • ConstantFieldExtension(F, C): RngDiff, Fld → RngDiff, Map
  • Example: Constant Field Extension of a Differential Field
  • Example: Constant Field Extension of a Laurent Series Ring
  • Completion(F, p): RngDiff, PlcFunElt → RngDiff, Map
  • Example: Laurent Series Ring by Completion
  • Example: Completion of an Elliptic Function Field
• Ring and Field Extensions
  • DifferentialRingExtension(L): RngDiffOpElt → RngDiff
  • DifferentialFieldExtension(L): RngDiffOpElt → RngDiff
  • Example: Differential Ring Extension by an Operator
  • Example: Differential Field Extension by an Operator
  • ext< F | f >: RngDiff, RngUPolElt → RngDiff
  • Example: Algebraic Extension of a Differential Field
  • ExponentialFieldExtension(F, f): RngDiff, RngDiffElt → RngDiff
  • LogarithmicFieldExtension(F, f): RngDiff, RngDiffElt → RngDiff
  • Example: Exponential and Logarithmic Extensions
  • PurelyRamifiedExtension(f): RngUPolElt[RngDiff] → RngDiff, Map
  • Example: Purely Ramified Extension of an Algebraic Differential Field
  • Example: Purely Ramified Extension of a Laurent Series Ring
  • Example: Purely Ramified Extension with an Inexact Derivation
• Ideals and Quotient Rings
  • Defining Ideals and Quotient Rings
    • DifferentialIdeal(L): [RngDiffElt] → RngMPol
    • QuotientRing(R, I): RngDiff, RngMPol → RngDiff, Map
    • Example: Differential Ideal and Quotient Ring
  • Boolean Operations on Ideals
    • IsDifferentialIdeal(R, I): RngDiff, RngMPol → BoolElt
• Wronskian Matrix
  • WronskianMatrix(L): [RngDiffElt] → AlgMatElt
  • WronskianDeterminant(L): [RngDiffElt] → RngDiffElt, AlgMatElt
  • Example: Wronskian Matrix and Determinant
• Differential Operator Rings
  • Creation
    • DifferentialOperatorRing(F): RngDiff → RngDiffOp
    • Example: Creating a Differential Operator Ring
    • AssignNames(~R, S): RngDiffOp, [MonStgElt]
  • Creation of Differential Operators
    • Name(R, i): RngDiffOp, RngIntElt → RngDiffElt
    • R . i: RngDiffOp, RngIntElt → RngDiffOpElt
    • R ! s: RngDiffOp, RngElt → RngDiffOpElt
    • Zero(R): RngDiffOp → RngDiffOpElt
    • One(R): RngDiffOp → RngDiffOpElt
    • Example: Coercion into an Operator Ring
• Structure Operations on Differential Operator Rings
  • Category and Parent
    • Category(R): RngDiffOp → RngDiffOp
    • Type(R): RngDiffOp → RngDiffOp
    • Parent(R): RngDiffOp → PowStr
  • Related Structures
    • BaseRing(R): RngDiffOp → Rng
    • CoefficientRing(R): RngDiffOp → Rng
    • ConstantRing(R): RngDiffOp → Rng
  • Derivation and Differential
    • Derivation(R): RngDiffOp → Map
    • Differential(R): RngDiffOp → DiffFunElt
    • Example: Related Structures of an Operator Ring
  • 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: Predicates for Operator Rings
    • Example: Predicates for Operator Rings over Series
  • Precision
    • RelativePrecisionOfDerivation(R): RngDiffOp → RngElt
    • Example: Relative Precision of an Operator Ring
• Element Operations on Differential Operators
  • Category and Parent
    • Category(L): RngDiffOpElt → RngDiffOpElt
    • Type(L): RngDiffOpElt → RngDiffOpElt
    • Parent(L): RngDiffOpElt → RngDiffOp
  • Arithmetic
    • s + t: RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • s + t: RngDiffOpElt, RngElt → RngDiffOpElt
    • s + t: RngElt, RngDiffOpElt → RngDiffOpElt
    • - s: RngDiffOpElt → RngDiffOpElt
    • s - t: RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • s - t: RngDiffOpElt, RngElt → RngDiffOpElt
    • s - t: RngElt, RngDiffOpElt → RngDiffOpElt
    • s * t: RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • s * t: RngDiffOpElt, RngElt → RngDiffOpElt
    • s * t: RngElt, RngDiffOpElt → RngDiffOpElt
    • s ^ n: RngDiffOpElt, RngIntElt → RngDiffElt
    • Example: Arithmetic of Differential Operators
  • 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
    • Coefficients(L): RngDiffOpElt → SeqEnum
    • Coefficient(L, i): RngDiffOpElt, RngIntElt → RngElt
    • LeadingCoefficient(L): RngDiffOpElt → RngElt
    • LeadingTerm(L): RngDiffOpElt → RngDiffOpElt
    • Terms(L): RngDiffOpElt → SeqEnum
    • Example: Coefficients and Terms of an Operator
  • Order and Degree
    • Order(L): RngDiffOpElt → RngIntElt
    • Degree(L): RngDiffOpElt → RngIntElt
    • WeakOrder(L): RngDiffOpElt → RngIntElt
    • WeakDegree(L): RngDiffOpElt → RngIntElt
    • Example: Order and Degree of Operators
  • Related Differential Operators
    • MonicDifferentialOperator(L): RngDiffOpElt → RngDiffOpElt
    • Adjoint(L): RngDiffOpElt → RngDiffOpElt
    • Translation(L, e): RngDiffOpElt, RngElt → RngDiffOpElt, Map
    • TruncateCoefficients(L): RngDiffOpElt → RngDiffOpElt
    • Example: Related Differential Operators
    • Example: Truncating Operator Coefficients
  • Application of Operators
    • Apply(L, f): RngDiffOpElt, RngElt → RngElt
    • L(f): RngElt, RngDiffOpElt → RngElt
    • f @ L: RngElt, RngDiffOpElt → RngElt
    • Example: Applying an Operator to a Function
• Related Maps
  • TranslationMap(R, e): RngDiffOp, RngElt → Map
  • LiftMap(m, R): Map, RngDiffOp → Map
  • Example: Maps between Operator Rings
• Changing Related Structures
  • ChangeDerivation(R, f): RngDiffOp, RngElt → RngDiffOp, Map
  • ChangeDifferential(R, df): RngDiffOp, DiffFunElt → RngDiffOp, Map
  • Example: Changing the Differential of an Operator Ring
  • ConstantFieldExtension(R, C): RngDiffOp, Fld → RngDiffOp, Map
  • PurelyRamifiedExtension(R,f): RngDiffOp, RngUPolElt → RngDiffOp, Map
  • Example: Purely Ramified Extension of an Operator Ring
  • 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: Changing Attributes of an Operator Ring over Series
  • Example: Completion of an Operator Ring
  • Example: Localization of an Operator Ring
• 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: Euclidean Right and Left Division
  • Greatest Common Right and Left Divisors
    • GreatestCommonRightDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • GCRD(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • ExtendedGreatestCommonRightDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt, RngDiffOpElt
    • GreatestCommonLeftDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • GCLD(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • ExtendedGreatestCommonLeftDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt, RngDiffOpElt
    • Example: Greatest Common Right and Left Divisors
  • Least Common Left Multiples
    • LeastCommonLeftMultiple(L): RngDiffOpElt → RngDiffOpElt
    • LeastCommonLeftMultiple(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • LCLM(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt
    • ExtendedLeastCommonLeftMultiple(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt, RngDiffOpElt
    • ExtendedLeastCommonLeftMultiple(S): [RngDiffOpElt] → RngDiffOpElt, SeqEnum
    • Example: Least Common Left Multiples
    • Example: Least Common Left Multiple of Conjugates
• Related Matrices
  • CompanionMatrix(L): RngDiffOpElt → AlgMatElt
  • Example: Companion Matrix of an Operator
• 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: Singular Places of an Operator
    • Example: Regular Singular Operators over Series
  • Indicial Polynomials
    • IndicialPolynomial(L, p): RngDiffOpElt, PlcFunElt → RngElt
    • Example: Indicial Polynomial of an Operator
• Rational Solutions
  • RationalSolutions(L): RngDiffOpElt → SeqEnum
  • HasRationalSolutions(L, g): RngDiffOpElt, RngElt → BoolElt, RngElt, SeqEnum
  • Example: Rational Solutions of an Operator
• Newton Polygons
  • NewtonPolygon(L): RngDiffOpElt → NwtnPgon, RingDiffOpElt
  • NewtonPolygon(L, p): RngDiffOpElt, PlcFunElt → NwtnPgon, RingDiffOpElt
  • NewtonPolynomial(F): NwtnPgonFace → RngUPolElt
  • NewtonPolynomials(L): RngDiffOpElt → SeqEnum, SeqEnum
  • Example: Newton Polygon of an Operator
  • Example: Newton Polygon at Infinity
  • Example: Newton Polygon over a Laurent Series Ring
• Symmetric Powers
  • SymmetricPower(L, m): RngDiffOpElt, RngIntElt → RngDiffOpElt
  • Example: Symmetric Powers of Operators
• Differential Operators of Algebraic Functions
  • DifferentialOperator(f): RngUPolElt → RngDiffOpElt
  • Example: Differential Operator of an Algebraic Function
• Factorisation of Operators over Differential Laurent Series Rings
  • Slope Valuation of an Operator
    • SlopeValuation(L,s): RngDiffOpElt, RngElt → FldRatElt
    • Example: Slope Valuation of an Operator
  • Coprime Index \(1\) and LCLM Factorisation
    • Factorisation(L): RngDiffOpElt → SeqEnum, SeqEnum
    • Factorization(L): RngDiffOpElt → SeqEnum, SeqEnum
    • Example: Operator with Infinitely Many Factorisations
    • Example: Comparing LCLM and Coprime Index 1 Factorisations
    • Example: Factorising a Product of First-Order Operators
    • Example: Operator the Factorisation Routine Cannot Split
    • Example: Unexpected Factorisation of an Operator
    • Example: Precision of Factors for a Rational Slope
  • Right Hand Factors of Operators
    • RightHandFactors(L): RngDiffOpElt → SeqEnum, [BoolElt]
    • Example: Right Hand Factors Matching an LCLM Factorisation
    • Example: Operator That Is Its Own Right Hand Factor
    • Example: Right Hand Factors Where Factorisation Fails
    • Example: Right Hand Factors for a Square Newton Polynomial
    • Example: Operators Sharing a Newton Polynomial
    • Example: Factorising an Operator of Order 9
• Bibliography