Creation
DifferentialRing(P, f, C) : Rng, Map, Rng -> RngDiff
Example RngDiff_diff-ring-create (H75E1)
RationalDifferentialField(C) : Fld -> RngDiff
Example RngDiff_rat-diff-field-create (H75E2)
DifferentialLaurentSeriesRing(C) : Fld -> RngDiff
Example RngDiff_diff-laur-ser-ring-create (H75E3)
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
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 (H75E4)
Example RngDiff_diff-laur-ser-related-structures (H75E5)
Derivation and Differential
Derivation(R) : RngDiff -> Map
Differential(F) : RngDiff -> DiffFunElt
Example RngDiff_diff-ring-derivation-differential (H75E6)
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 (H75E7)
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 (H75E8)
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 (H75E9)
Example RngDiff_Example-coefficients-terms-differential-series (H75E10)
Conjugates, Norm and Trace
MinimalPolynomial(s) : RngDiffElt -> RngUPolElt
Example RngDiff_Example-minimal-polynomial-differential-rings (H75E11)
Functions on Elements
Derivative(s) : RngDiffElt -> RngDiffElt
Example RngDiff_derivative (H75E12)
Changing Related Structures
ChangeDerivation(R, f) : RngDiff, RngElt -> RngDiff, Map
Example RngDiff_diff-ring-change-derivation (H75E13)
ConstantFieldExtension(F, C) : RngDiff, Fld -> RngDiff, Map
Example RngDiff_diff-ring-constant-field-extension (H75E14)
Example RngDiff_diff-ring-constant-field-extension-series (H75E15)
Completion(F, p) : RngDiff, PlcFunElt -> RngDiff, Map
Example RngDiff_diff-ring-completion-create (H75E16)
Example RngDiff_diff-ring-completion-elliptic (H75E17)
Ring and Field Extensions
DifferentialRingExtension(L) : RngDiffOpElt -> RngDiff
DifferentialFieldExtension(L) : RngDiffOpElt, -> RngDiff
Example RngDiff_diff-ring-ext-operator (H75E18)
Example RngDiff_diff-field-ext-operator (H75E19)
ext< F | f > : RngDiff, RngUPolElt -> RngDiff
Example RngDiff_diff-field-ext-ext (H75E20)
ExponentialFieldExtension(F, f) : RngDiff, RngDiffElt -> RngDiff
LogarithmicFieldExtension(F, f) : RngDiff, RngDiffElt -> RngDiff
Example RngDiff_diff-field-ext-exp-log (H75E21)
Defining Ideals and Quotient Rings
DifferentialIdeal(L) : [RngDiffElt] -> RngMPol
QuotientRing(R, I) : RngDiff, RngMPol -> RngDiff, Map
Example RngDiff_diff-ideal-quotient (H75E22)
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 (H75E23)
Creation
DifferentialOperatorRing(F) : RngDiff -> RngDiffOp
Example RngDiff_create-diff-op-ring (H75E24)
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 (H75E25)
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 (H75E26)
Predicates and Booleans
R eq F : RngDiffOp, RngDiffOp -> BoolElt
IsIdentical(R, F) : RngDiffOp, RngDiffOp -> BoolElt
IsDifferentialOperatorRing(R) : . -> BoolElt
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 (H75E27)
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 (H75E28)
Order and Degree
Order(L) : RngDiffOpElt -> RngIntElt
WeakOrder(L) : RngDiffOpElt -> RngIntElt
Example RngDiff_example-booleans-degrees-diff-ops (H75E29)
Related Differential Operators
MonicDifferentialOperator(L) : RngDiffOpElt -> RngDiffOpElt
Adjoint(L) : RngDiffOpElt -> RngDiffOpElt
Translation(L, e) : RngDiffOpElt, RngElt -> RngDiffOpElt, Map
Example RngDiff_related-diff-op (H75E30)
Application of Operators
Apply(L, f) : RngDiffOpElt, RngElt -> RngElt
Example RngDiff_example-apply (H75E31)
Related Maps
TranslationMap(R, e) : RngDiffOp, RngElt -> Map
LiftMap(m, R) : Map, RngDiffOp -> Map
Example RngDiff_example-maps-between-diff-op-rings (H75E32)
Changing Related Structures
ChangeDerivation(R, f) : RngDiffOp, RngElt -> RngDiffOp, Map
ConstantFieldExtension(R, C) : RngDiffOp,Fld -> RngDiffOp, Map
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 (H75E33)
Example RngDiff_example-diff-op-ring-completion (H75E34)
Example RngDiff_example-diff-op-ring-localization (H75E35)
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 (H75E36)
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 (H75E37)
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 (H75E38)
Example RngDiff_example-LCLM-conjugates (H75E39)
Related Matrices
CompanionMatrix(L) : RngDiffOpElt -> AlgMatElt
Example RngDiff_example-companion-matrix (H75E40)
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 (H75E41)
Example RngDiff_example-regular-singular-DLSR (H75E42)
Indicial Polynomials
IndicialPolynomial(L, p) : RngDiffOpElt, PlcFunElt -> RngElt
Example RngDiff_example-indicial-pol (H75E43)
Rational Solutions
RationalSolutions(L) : RngDiffOpElt, -> SeqEnum
HasRationalSolutions(L, g) : RngDiffOpElt, RngElt -> BoolElt, RngElt, SeqEnum
Example RngDiff_example-rational-sols (H75E44)
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 (H75E45)
Example RngDiff_example-newton-pgns-2 (H75E46)
Example RngDiff_example-newton-pgns-3 (H75E47)
Symmetric Powers
SymmetricPower(L, m) : RngDiffOpElt, RngIntElt -> RngDiffOpElt
Example RngDiff_example-symmetric-power (H75E48)
Differential Operators of Algebraic Functions
DifferentialOperator(f) : RngUPolElt -> RngDiffOpElt