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)
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)
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
Mon Dec 17 14:40:36 EST 2012