# Differential Rings

- [Introduction](introduction.md)

- [Differential Rings and Fields](creation-diff-rings.md)

  - [Creation](creation-diff-rings.md#creation)

    - [`DifferentialRing(P, f, C): Rng, Map, Rng → RngDiff`](creation-diff-rings.md#function-differentialring-rng-map-rng)

    - [`Example: Diff Ring Create`](creation-diff-rings.md#example-ex-dba016)

    - [`RationalDifferentialField(C): Fld → RngDiff`](creation-diff-rings.md#function-rationaldifferentialfield-fld)

    - [`Example: Rat Diff Field Create`](creation-diff-rings.md#example-ex-c5e585)

    - [`DifferentialLaurentSeriesRing(C): Fld → RngDiff`](creation-diff-rings.md#function-differentiallaurentseriesring-fld)

    - [`Example: Diff Laur Ser Ring Create`](creation-diff-rings.md#example-ex-d698ec)

    - [`RingOfFractions(R): RngDiff → RngDiff, Map`](creation-diff-rings.md#function-ringoffractions-rngdiff)

    - [`FieldOfFractions(R): RngDiff → RngDiff, Map`](creation-diff-rings.md#function-fieldoffractions-rngdiff)

    - [`AssignNames(~R, S): RngDiff, [MonStgElt]`](creation-diff-rings.md#function-assignnames-rngdiff-monstgelt-ref)

  - [Creation of Differential Ring Elements](creation-diff-rings.md#creation-of-differential-ring-elements)

    - [`Name(R, i): RngDiff, RngIntElt → RngDiffElt`](creation-diff-rings.md#function-name-rngdiff-rngintelt)

    - [`R . i: RngDiff, RngIntElt → RngDiffElt`](creation-diff-rings.md#operation-operation-rngdiff-rngintelt-rngdiffelt)

    - [`R ! s: RngDiff, RngElt → RngDiffElt`](creation-diff-rings.md#operation-op-rngdiff-rngelt)

    - [`Zero(R): RngDiff → RngDiffElt`](creation-diff-rings.md#function-zero-rngdiff)

    - [`One(R): RngDiff → RngDiffElt`](creation-diff-rings.md#function-one-rngdiff)

    - [`Identity(R): RngDiff → RngDiffElt`](creation-diff-rings.md#function-identity-rngdiff)

    - [`SeparatingElement(F): RngDiff → RngDiffElt`](creation-diff-rings.md#function-separatingelement-rngdiff)

    - [`Example: Diff Ring Element Creation`](creation-diff-rings.md#example-ex-284f4d)

- [Structure Operations on Differential Rings](structure-operations-diff-rings.md)

  - [Category and Parent](structure-operations-diff-rings.md#category-and-parent)

    - [`Category(R): RngDiff → RngDiff`](structure-operations-diff-rings.md#function-category-rngdiff)

    - [`Type(R): RngDiff → RngDiff`](structure-operations-diff-rings.md#function-type-rngdiff)

    - [`Parent(R): RngDiff → PowStr`](structure-operations-diff-rings.md#function-parent-rngdiff)

  - [Related Structures](structure-operations-diff-rings.md#related-structures)

    - [`UnderlyingRing(R): RngDiff → Rng`](structure-operations-diff-rings.md#function-underlyingring-rngdiff)

    - [`UnderlyingField(R): RngDiff → Rng`](structure-operations-diff-rings.md#function-underlyingfield-rngdiff)

    - [`BaseRing(R): RngDiff → Rng`](structure-operations-diff-rings.md#function-basering-rngdiff)

    - [`BaseField(R): RngDiff → Rng`](structure-operations-diff-rings.md#function-basefield-rngdiff)

    - [`ConstantRing(R): RngDiff → Rng`](structure-operations-diff-rings.md#function-constantring-rngdiff)

    - [`ConstantField(R): RngDiff → Rng`](structure-operations-diff-rings.md#function-constantfield-rngdiff)

    - [`ExactConstantField(F): RngDiff → RngDiff, Map`](structure-operations-diff-rings.md#function-exactconstantfield-rngdiff)

    - [`Generators(R): RngDiff → SeqEnum`](structure-operations-diff-rings.md#function-generators-rngdiff)

    - [`Example: Diff Ring Related Structures`](structure-operations-diff-rings.md#example-ex-64f77a)

    - [`Example: Diff Laur Ser Related Structures`](structure-operations-diff-rings.md#example-ex-326abb)

  - [Derivation and Differential](structure-operations-diff-rings.md#derivation-and-differential)

    - [`Derivation(R): RngDiff → Map`](structure-operations-diff-rings.md#function-derivation-rngdiff)

    - [`Differential(F): RngDiff → DiffFunElt`](structure-operations-diff-rings.md#function-differential-rngdiff)

    - [`Example: Diff Ring Derivation Differential`](structure-operations-diff-rings.md#example-ex-627c5f)

  - [Numerical Invariants](structure-operations-diff-rings.md#numerical-invariants)

    - [`Ngens(R): RngDiff → RngIntElt`](structure-operations-diff-rings.md#function-ngens-rngdiff)

  - [Predicates and Booleans](structure-operations-diff-rings.md#predicates-and-booleans)

    - [`R eq F: RngDiff, RngDiff → BoolElt`](structure-operations-diff-rings.md#operation-op-eq-rngdiff-rngdiff)

    - [`IsIdentical(R, F): RngDiff, RngDiff → BoolElt`](structure-operations-diff-rings.md#function-isidentical-rngdiff-rngdiff)

    - [`IsDomain(R): RngDiff → BoolElt`](structure-operations-diff-rings.md#function-isdomain-rngdiff)

    - [`IsField(R): RngDiff → BoolElt`](structure-operations-diff-rings.md#function-isfield-rngdiff)

    - [`IsDifferentialField(R): Rng → BoolElt`](structure-operations-diff-rings.md#function-isdifferentialfield-rng)

    - [`IsAlgebraicDifferentialField(R): Rng → BoolElt`](structure-operations-diff-rings.md#function-isalgebraicdifferentialfield-rng)

    - [`IsDifferentialSeriesRing(R): Rng → BoolElt`](structure-operations-diff-rings.md#function-isdifferentialseriesring-rng)

    - [`IsDifferentialLaurentSeriesRing(R): Rng → BoolElt`](structure-operations-diff-rings.md#function-isdifferentiallaurentseriesring-rng)

    - [`Example: Diff Rings Booleans`](structure-operations-diff-rings.md#example-ex-913ce7)

    - [`HasProjectiveDerivation(F): RngDiff → BoolElt`](structure-operations-diff-rings.md#function-hasprojectivederivation-rngdiff)

    - [`HasZeroDerivation(F): RngDiff → BoolElt`](structure-operations-diff-rings.md#function-haszeroderivation-rngdiff)

    - [`Example: Diff Rings Booleans Derivation`](structure-operations-diff-rings.md#example-ex-a3014d)

  - [Precision](structure-operations-diff-rings.md#precision)

    - [`RelativePrecision(F): RngDiff → RngElt`](structure-operations-diff-rings.md#function-relativeprecision-rngdiff)

    - [`RelativePrecisionOfDerivation(F): RngDiff → RngElt`](structure-operations-diff-rings.md#function-relativeprecisionofderivation-rngdiff)

    - [`Example: Diff Rings Relative Precision`](structure-operations-diff-rings.md#example-ex-4674cf)

    - [`Example Differential Ring Relative Precision Derivation`](structure-operations-diff-rings.md#example-ex-617005)

    - [`ChangePrecision(F, p): RngDiff, RngElt → RngDiff, Map`](structure-operations-diff-rings.md#function-changeprecision-rngdiff-rngelt)

    - [`Example: Diff Rings Change Precision`](structure-operations-diff-rings.md#example-ex-d18d60)

- [Element Operations on Differential Ring Elements](element-operations.md)

  - [Category and Parent](element-operations.md#category-and-parent)

    - [`Category(s): RngDiffElt → RngDiffElt`](element-operations.md#function-category-rngdiffelt)

    - [`Type(s): RngDiffElt → RngDiffElt`](element-operations.md#function-type-rngdiffelt)

    - [`Parent(s): RngDiffElt → RngDiff`](element-operations.md#function-parent-rngdiffelt)

  - [Arithmetic](element-operations.md#arithmetic)

    - [`s + t: RngDiffElt, RngDiffElt → RngDiffElt`](element-operations.md#operation-op-plus-rngdiffelt-rngdiffelt)

    - [`- s: RngDiffElt → RngDiffElt`](element-operations.md#operation-operation-rngdiffelt-rngdiffelt)

    - [`s - t: RngDiffElt, RngDiffElt → RngDiffElt`](element-operations.md#operation-op-minus-rngdiffelt-rngdiffelt)

    - [`s * t: RngDiffElt, RngDiffElt → RngDiffElt`](element-operations.md#operation-op-times-rngdiffelt-rngdiffelt)

    - [`s ^ n: RngDiffElt, RngIntElt → RngDiffElt`](element-operations.md#operation-op-pow-rngdiffelt-rngintelt)

    - [`s div t: RngDiffElt, RngDiffElt → RngDiffElt`](element-operations.md#operation-operation-div-rngdiffelt-rngdiffelt-rngdiffelt)

    - [`s / t: RngDiffElt, RngDiffElt → RngDiffElt`](element-operations.md#operation-op-div-rngdiffelt-rngdiffelt)

  - [Predicates and Booleans](element-operations.md#predicates-and-booleans)

    - [`s eq t: RngDiffElt, RngDiffElt → BoolElt`](element-operations.md#operation-op-eq-rngdiffelt-rngdiffelt)

    - [`IsZero(s): RngDiffElt → BoolElt`](element-operations.md#function-iszero-rngdiffelt)

    - [`IsOne(s): RngDiffElt → BoolElt`](element-operations.md#function-isone-rngdiffelt)

    - [`IsWeaklyEqual(s, t): RngDiffElt, RngDiffElt → BoolElt`](element-operations.md#function-isweaklyequal-rngdiffelt-rngdiffelt)

    - [`IsWeaklyZero(s): RngDiffElt → BoolElt`](element-operations.md#function-isweaklyzero-rngdiffelt)

    - [`IsOrderTerm(s): RngDiffElt → BoolElt`](element-operations.md#function-isorderterm-rngdiffelt)

    - [`IsOrderTerm(s): RngSerElt → BoolElt`](element-operations.md#function-isorderterm-rngserelt)

    - [`Example: Diff Ring Elts Booleans`](element-operations.md#example-ex-61c4b6)

  - [Coefficients and Terms](element-operations.md#coefficients-and-terms)

    - [`O(s): RngDiffElt → RngDiffElt`](element-operations.md#function-o-rngdiffelt)

    - [`Truncate(s): RngDiffElt → RngDiffElt`](element-operations.md#function-truncate-rngdiffelt)

    - [`Eltseq(s): RngDiffElt → SeqEnum`](element-operations.md#function-eltseq-rngdiffelt)

    - [`Exponents(s): RngDiffElt → SeqEnum`](element-operations.md#function-exponents-rngdiffelt)

    - [`Exponents(s): RngSerElt → SeqEnum`](element-operations.md#function-exponents-rngserelt)

    - [`Example Coefficients Terms Algebraic Differential Elements`](element-operations.md#example-ex-18d9d7)

    - [`Example Coefficients Terms Differential Series`](element-operations.md#example-ex-4eef4b)

  - [Conjugates, Norm and Trace](element-operations.md#conjugates-norm-and-trace)

    - [`MinimalPolynomial(s): RngDiffElt → RngUPolElt`](element-operations.md#function-minimalpolynomial-rngdiffelt)

    - [`Example Minimal Polynomial Differential Rings`](element-operations.md#example-ex-ecfdbc)

  - [Derivatives and Differentials](element-operations.md#derivatives-and-differentials)

    - [`Derivative(s): RngDiffElt → RngDiffElt`](element-operations.md#function-derivative-rngdiffelt)

    - [`Differential(s): RngDiffElt → RngDiffElt`](element-operations.md#function-differential-rngdiffelt)

    - [`Example: Derivative Differential Diff Ring Elements`](element-operations.md#example-ex-396a0f)

- [Changing Related Structures](changing-attributes-diff-rings.md)

  - [`ChangeDerivation(R, f): RngDiff, RngElt → RngDiff, Map`](changing-attributes-diff-rings.md#function-changederivation-rngdiff-rngelt)

  - [`Example: Diff Ring Change Derivation`](changing-attributes-diff-rings.md#example-ex-271826)

  - [`ChangeDifferential(F, df): RngDiff, DiffFunElt → RngDiff, Map`](changing-attributes-diff-rings.md#function-changedifferential-rngdiff-difffunelt)

  - [`Example: Diff Ring Change Differential`](changing-attributes-diff-rings.md#example-ex-7c6694)

  - [`ConstantFieldExtension(F, C): RngDiff, Fld → RngDiff, Map`](changing-attributes-diff-rings.md#function-constantfieldextension-rngdiff-fld)

  - [`Example: Diff Ring Constant Field Extension`](changing-attributes-diff-rings.md#example-ex-17cbbe)

  - [`Example: Diff Ring Constant Field Extension Series`](changing-attributes-diff-rings.md#example-ex-a878b1)

  - [`Completion(F, p): RngDiff, PlcFunElt → RngDiff, Map`](changing-attributes-diff-rings.md#function-completion-rngdiff-plcfunelt)

  - [`Example: Diff Ring Completion Create`](changing-attributes-diff-rings.md#example-ex-5fcb44)

  - [`Example: Diff Ring Completion Elliptic`](changing-attributes-diff-rings.md#example-ex-512dde)

- [Ring and Field Extensions](extensions-ring-diff.md)

  - [`DifferentialRingExtension(L): RngDiffOpElt → RngDiff`](extensions-ring-diff.md#function-differentialringextension-rngdiffopelt)

  - [`DifferentialFieldExtension(L): RngDiffOpElt → RngDiff`](extensions-ring-diff.md#function-differentialfieldextension-rngdiffopelt)

  - [`Example: Diff Ring Ext Operator`](extensions-ring-diff.md#example-ex-434a22)

  - [`Example: Diff Field Ext Operator`](extensions-ring-diff.md#example-ex-5ef2f3)

  - [`ext< F | f >: RngDiff, RngUPolElt → RngDiff`](extensions-ring-diff.md#constructor-constructor-ext-rngdiff-rngupolelt-rngdiff)

  - [`Example: Diff Field Ext Ext`](extensions-ring-diff.md#example-ex-8a3a0e)

  - [`ExponentialFieldExtension(F, f): RngDiff, RngDiffElt → RngDiff`](extensions-ring-diff.md#function-exponentialfieldextension-rngdiff-rngdiffelt)

  - [`LogarithmicFieldExtension(F, f): RngDiff, RngDiffElt → RngDiff`](extensions-ring-diff.md#function-logarithmicfieldextension-rngdiff-rngdiffelt)

  - [`Example: Diff Field Ext Exp Log`](extensions-ring-diff.md#example-ex-0b50af)

  - [`PurelyRamifiedExtension(f): RngUPolElt[RngDiff] → RngDiff, Map`](extensions-ring-diff.md#function-purelyramifiedextension-rngupolelt-rngdiff)

  - [`Example: Diff Field Purely Ramified Ext ADF`](extensions-ring-diff.md#example-ex-838a6e)

  - [`Example: Diff Field Purely Ramified Ext DLSR 1`](extensions-ring-diff.md#example-ex-efad09)

  - [`Example: Diff Field Purely Ramified Ext DLSR 2`](extensions-ring-diff.md#example-ex-da702b)

- [Ideals and Quotient Rings](ideal-quotient-ring.md)

  - [Defining Ideals and Quotient Rings](ideal-quotient-ring.md#defining-ideals-and-quotient-rings)

    - [`DifferentialIdeal(L): [RngDiffElt] → RngMPol`](ideal-quotient-ring.md#function-differentialideal-rngdiffelt)

    - [`QuotientRing(R, I): RngDiff, RngMPol → RngDiff, Map`](ideal-quotient-ring.md#function-quotientring-rngdiff-rngmpol)

    - [`Example: Diff Ideal Quotient`](ideal-quotient-ring.md#example-ex-effe4a)

  - [Boolean Operations on Ideals](ideal-quotient-ring.md#boolean-operations-on-ideals)

    - [`IsDifferentialIdeal(R, I): RngDiff, RngMPol → BoolElt`](ideal-quotient-ring.md#function-isdifferentialideal-rngdiff-rngmpol)

- [Wronskian Matrix](wronskians.md)

  - [`WronskianMatrix(L): [RngDiffElt] → AlgMatElt`](wronskians.md#function-wronskianmatrix-rngdiffelt)

  - [`WronskianDeterminant(L): [RngDiffElt] → RngDiffElt, AlgMatElt`](wronskians.md#function-wronskiandeterminant-rngdiffelt)

  - [`Example: Wronskian Mat Det`](wronskians.md#example-ex-95dc85)

- [Differential Operator Rings](creation-diff-op-rings.md)

  - [Creation](creation-diff-op-rings.md#creation)

    - [`DifferentialOperatorRing(F): RngDiff → RngDiffOp`](creation-diff-op-rings.md#function-differentialoperatorring-rngdiff)

    - [`Example: Create Diff Op Ring`](creation-diff-op-rings.md#example-ex-42ff41)

    - [`AssignNames(~R, S): RngDiffOp, [MonStgElt]`](creation-diff-op-rings.md#function-assignnames-rngdiffop-monstgelt-ref)

  - [Creation of Differential Operators](creation-diff-op-rings.md#creation-of-differential-operators)

    - [`Name(R, i): RngDiffOp, RngIntElt → RngDiffElt`](creation-diff-op-rings.md#function-name-rngdiffop-rngintelt)

    - [`R . i: RngDiffOp, RngIntElt → RngDiffOpElt`](creation-diff-op-rings.md#operation-operation-rngdiffop-rngintelt-rngdiffopelt)

    - [`R ! s: RngDiffOp, RngElt → RngDiffOpElt`](creation-diff-op-rings.md#operation-op-rngdiffop-rngelt)

    - [`Zero(R): RngDiffOp → RngDiffOpElt`](creation-diff-op-rings.md#function-zero-rngdiffop)

    - [`One(R): RngDiffOp → RngDiffOpElt`](creation-diff-op-rings.md#function-one-rngdiffop)

    - [`Example Diff Op Coercion`](creation-diff-op-rings.md#example-ex-facb79)

- [Structure Operations on Differential Operator Rings](structure-operations-diff-op-rings.md)

  - [Category and Parent](structure-operations-diff-op-rings.md#category-and-parent)

    - [`Category(R): RngDiffOp → RngDiffOp`](structure-operations-diff-op-rings.md#function-category-rngdiffop)

    - [`Type(R): RngDiffOp → RngDiffOp`](structure-operations-diff-op-rings.md#function-type-rngdiffop)

    - [`Parent(R): RngDiffOp → PowStr`](structure-operations-diff-op-rings.md#function-parent-rngdiffop)

  - [Related Structures](structure-operations-diff-op-rings.md#related-structures)

    - [`BaseRing(R): RngDiffOp → Rng`](structure-operations-diff-op-rings.md#function-basering-rngdiffop)

    - [`CoefficientRing(R): RngDiffOp → Rng`](structure-operations-diff-op-rings.md#function-coefficientring-rngdiffop)

    - [`ConstantRing(R): RngDiffOp → Rng`](structure-operations-diff-op-rings.md#function-constantring-rngdiffop)

  - [Derivation and Differential](structure-operations-diff-op-rings.md#derivation-and-differential)

    - [`Derivation(R): RngDiffOp → Map`](structure-operations-diff-op-rings.md#function-derivation-rngdiffop)

    - [`Differential(R): RngDiffOp → DiffFunElt`](structure-operations-diff-op-rings.md#function-differential-rngdiffop)

    - [`Example: Diff Op Ring Related Structures`](structure-operations-diff-op-rings.md#example-ex-fcdd49)

  - [Predicates and Booleans](structure-operations-diff-op-rings.md#predicates-and-booleans)

    - [`R eq F: RngDiffOp, RngDiffOp → BoolElt`](structure-operations-diff-op-rings.md#operation-op-eq-rngdiffop-rngdiffop)

    - [`IsIdentical(R, F): RngDiffOp, RngDiffOp → BoolElt`](structure-operations-diff-op-rings.md#function-isidentical-rngdiffop-rngdiffop)

    - [`IsDifferentialOperatorRing(R): . → BoolElt`](structure-operations-diff-op-rings.md#function-isdifferentialoperatorring)

    - [`HasProjectiveDerivation(R): RngDiffOp → BoolElt`](structure-operations-diff-op-rings.md#function-hasprojectivederivation-rngdiffop)

    - [`HasZeroDerivation(R): RngDiffOp → BoolElt`](structure-operations-diff-op-rings.md#function-haszeroderivation-rngdiffop)

    - [`Example: Diff Op Ring Booleans`](structure-operations-diff-op-rings.md#example-ex-74f01d)

    - [`Example: Diff Op Ring Booleans LSR`](structure-operations-diff-op-rings.md#example-ex-f354a2)

  - [Precision](structure-operations-diff-op-rings.md#precision)

    - [`RelativePrecisionOfDerivation(R): RngDiffOp → RngElt`](structure-operations-diff-op-rings.md#function-relativeprecisionofderivation-rngdiffop)

    - [`Example: Diff Op Rings Relative Precision`](structure-operations-diff-op-rings.md#example-ex-1465b5)

- [Element Operations on Differential Operators](element-operations-diff-op-rings.md)

  - [Category and Parent](element-operations-diff-op-rings.md#category-and-parent)

    - [`Category(L): RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#function-category-rngdiffopelt)

    - [`Type(L): RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#function-type-rngdiffopelt)

    - [`Parent(L): RngDiffOpElt → RngDiffOp`](element-operations-diff-op-rings.md#function-parent-rngdiffopelt)

  - [Arithmetic](element-operations-diff-op-rings.md#arithmetic)

    - [`s + t: RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-plus-rngdiffopelt-rngdiffopelt)

    - [`s + t: RngDiffOpElt, RngElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-plus-rngdiffopelt-rngelt)

    - [`s + t: RngElt, RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-plus-rngelt-rngdiffopelt)

    - [`- s: RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-operation-rngdiffopelt-rngdiffopelt)

    - [`s - t: RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-minus-rngdiffopelt-rngdiffopelt)

    - [`s - t: RngDiffOpElt, RngElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-minus-rngdiffopelt-rngelt)

    - [`s - t: RngElt, RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-minus-rngelt-rngdiffopelt)

    - [`s * t: RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-times-rngdiffopelt-rngdiffopelt)

    - [`s * t: RngDiffOpElt, RngElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-times-rngdiffopelt-rngelt)

    - [`s * t: RngElt, RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#operation-op-times-rngelt-rngdiffopelt)

    - [`s ^ n: RngDiffOpElt, RngIntElt → RngDiffElt`](element-operations-diff-op-rings.md#operation-op-pow-rngdiffopelt-rngintelt)

    - [`Example: Diff Op Arithmetic`](element-operations-diff-op-rings.md#example-ex-a0ba79)

  - [Predicates and Booleans](element-operations-diff-op-rings.md#predicates-and-booleans)

    - [`s eq t: RngDiffOpElt, RngDiffOpElt → BoolElt`](element-operations-diff-op-rings.md#operation-op-eq-rngdiffopelt-rngdiffopelt)

    - [`IsZero(L): RngDiffOpElt → BoolElt`](element-operations-diff-op-rings.md#function-iszero-rngdiffopelt)

    - [`IsOne(L): RngDiffOpElt → BoolElt`](element-operations-diff-op-rings.md#function-isone-rngdiffopelt)

    - [`IsMonic(L): RngDiffOpElt → BoolElt`](element-operations-diff-op-rings.md#function-ismonic-rngdiffopelt)

    - [`IsWeaklyEqual(L, P): RngDiffOpElt, RngDiffOpElt → BoolElt`](element-operations-diff-op-rings.md#function-isweaklyequal-rngdiffopelt-rngdiffopelt)

    - [`IsWeaklyZero(L): RngDiffOpElt → BoolElt`](element-operations-diff-op-rings.md#function-isweaklyzero-rngdiffopelt)

    - [`IsWeaklyMonic(L): RngDiffOpElt → BoolElt`](element-operations-diff-op-rings.md#function-isweaklymonic-rngdiffopelt)

  - [Coefficients and Terms](element-operations-diff-op-rings.md#coefficients-and-terms)

    - [`Eltseq(L): RngDiffOpElt → SeqEnum`](element-operations-diff-op-rings.md#function-eltseq-rngdiffopelt)

    - [`Coefficients(L): RngDiffOpElt → SeqEnum`](element-operations-diff-op-rings.md#function-coefficients-rngdiffopelt)

    - [`Coefficient(L, i): RngDiffOpElt, RngIntElt → RngElt`](element-operations-diff-op-rings.md#function-coefficient-rngdiffopelt-rngintelt)

    - [`LeadingCoefficient(L): RngDiffOpElt → RngElt`](element-operations-diff-op-rings.md#function-leadingcoefficient-rngdiffopelt)

    - [`LeadingTerm(L): RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#function-leadingterm-rngdiffopelt)

    - [`Terms(L): RngDiffOpElt → SeqEnum`](element-operations-diff-op-rings.md#function-terms-rngdiffopelt)

    - [`Example: Diff Op Coeff Terms`](element-operations-diff-op-rings.md#example-ex-d50a45)

  - [Order and Degree](element-operations-diff-op-rings.md#order-and-degree)

    - [`Order(L): RngDiffOpElt → RngIntElt`](element-operations-diff-op-rings.md#function-order-rngdiffopelt)

    - [`Degree(L): RngDiffOpElt → RngIntElt`](element-operations-diff-op-rings.md#function-degree-rngdiffopelt)

    - [`WeakOrder(L): RngDiffOpElt → RngIntElt`](element-operations-diff-op-rings.md#function-weakorder-rngdiffopelt)

    - [`WeakDegree(L): RngDiffOpElt → RngIntElt`](element-operations-diff-op-rings.md#function-weakdegree-rngdiffopelt)

    - [`Example Booleans Degrees Diff Ops`](element-operations-diff-op-rings.md#example-ex-873e6e)

  - [Related Differential Operators](element-operations-diff-op-rings.md#related-differential-operators)

    - [`MonicDifferentialOperator(L): RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#function-monicdifferentialoperator-rngdiffopelt)

    - [`Adjoint(L): RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#function-adjoint-rngdiffopelt)

    - [`Translation(L, e): RngDiffOpElt, RngElt → RngDiffOpElt, Map`](element-operations-diff-op-rings.md#function-translation-rngdiffopelt-rngelt)

    - [`TruncateCoefficients(L): RngDiffOpElt → RngDiffOpElt`](element-operations-diff-op-rings.md#function-truncatecoefficients-rngdiffopelt)

    - [`Example: Related Diff Op`](element-operations-diff-op-rings.md#example-ex-978632)

    - [`Example: Related Diff Op Truncate Coefficients`](element-operations-diff-op-rings.md#example-ex-32cf07)

  - [Application of Operators](element-operations-diff-op-rings.md#application-of-operators)

    - [`Apply(L, f): RngDiffOpElt, RngElt → RngElt`](element-operations-diff-op-rings.md#function-apply-rngdiffopelt-rngelt)

    - [`L(f): RngElt, RngDiffOpElt → RngElt`](element-operations-diff-op-rings.md#function-l-rngelt-rngdiffopelt)

    - [`f @ L: RngElt, RngDiffOpElt → RngElt`](element-operations-diff-op-rings.md#operation-operation-rngelt-rngdiffopelt-rngelt)

    - [`Example Apply`](element-operations-diff-op-rings.md#example-ex-1a4536)

- [Related Maps](related-maps-diff-op-rings.md)

  - [`TranslationMap(R, e): RngDiffOp, RngElt → Map`](related-maps-diff-op-rings.md#function-translationmap-rngdiffop-rngelt)

  - [`LiftMap(m, R): Map, RngDiffOp → Map`](related-maps-diff-op-rings.md#function-liftmap-map-rngdiffop)

  - [`Example Maps Between Diff Op Rings`](related-maps-diff-op-rings.md#example-ex-943ab0)

- [Changing Related Structures](changing-attributes-diff-op-rings.md)

  - [`ChangeDerivation(R, f): RngDiffOp, RngElt → RngDiffOp, Map`](changing-attributes-diff-op-rings.md#function-changederivation-rngdiffop-rngelt)

  - [`ChangeDifferential(R, df): RngDiffOp, DiffFunElt → RngDiffOp, Map`](changing-attributes-diff-op-rings.md#function-changedifferential-rngdiffop-difffunelt)

  - [`Example Diff Op Ring Changing Differential`](changing-attributes-diff-op-rings.md#example-ex-29b98d)

  - [`ConstantFieldExtension(R, C): RngDiffOp, Fld → RngDiffOp, Map`](changing-attributes-diff-op-rings.md#function-constantfieldextension-rngdiffop-fld)

  - [`PurelyRamifiedExtension(R,f): RngDiffOp, RngUPolElt → RngDiffOp, Map`](changing-attributes-diff-op-rings.md#function-purelyramifiedextension-rngdiffop-rngupolelt)

  - [`Example: Diff Op Ring Purely Ramified Extension DLSR`](changing-attributes-diff-op-rings.md#example-ex-43ec30)

  - [`Completion(R, p): RngDiffOp, PlcFunElt → RngDiffOp, Map`](changing-attributes-diff-op-rings.md#function-completion-rngdiffop-plcfunelt)

  - [`Localization(R, p): RngDiffOp, PlcFunElt → RngDiffOp, Map, PlcFunElt`](changing-attributes-diff-op-rings.md#function-localization-rngdiffop-plcfunelt)

  - [`Localization(L, p): RngDiffOpElt, PlcFunElt → RngDiffOpElt, Map, PlcFunElt`](changing-attributes-diff-op-rings.md#function-localization-rngdiffopelt-plcfunelt)

  - [`Localization(R): RngDiffOp → RngDiffOp, Map`](changing-attributes-diff-op-rings.md#function-localization-rngdiffop)

  - [`Localization(L): RngDiffOpElt → RngDiffOpElt, Map`](changing-attributes-diff-op-rings.md#function-localization-rngdiffopelt)

  - [`Example Diff Op Ring Changing Attributes DLSR`](changing-attributes-diff-op-rings.md#example-ex-7675d6)

  - [`Example Diff Op Ring Completion`](changing-attributes-diff-op-rings.md#example-ex-397606)

  - [`Example Diff Op Ring Localization`](changing-attributes-diff-op-rings.md#example-ex-885bd3)

- [Euclidean Algorithms, GCDs and LCMs](euclidean-algorithms.md)

  - [Euclidean Right and Left Division](euclidean-algorithms.md#euclidean-right-and-left-division)

    - [`EuclideanRightDivision(N, D): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt`](euclidean-algorithms.md#function-euclideanrightdivision-rngdiffopelt-rngdiffopelt)

    - [`EuclideanLeftDivision(D, N): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt`](euclidean-algorithms.md#function-euclideanleftdivision-rngdiffopelt-rngdiffopelt)

    - [`Example Eucl Alg`](euclidean-algorithms.md#example-ex-872ed1)

  - [Greatest Common Right and Left Divisors](euclidean-algorithms.md#greatest-common-right-and-left-divisors)

    - [`GreatestCommonRightDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](euclidean-algorithms.md#function-greatestcommonrightdivisor-rngdiffopelt-rngdiffopelt)

    - [`GCRD(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](euclidean-algorithms.md#function-gcrd-rngdiffopelt-rngdiffopelt)

    - [`ExtendedGreatestCommonRightDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt, RngDiffOpElt`](euclidean-algorithms.md#function-extendedgreatestcommonrightdivisor-rngdiffopelt-rngdiffopelt)

    - [`GreatestCommonLeftDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](euclidean-algorithms.md#function-greatestcommonleftdivisor-rngdiffopelt-rngdiffopelt)

    - [`GCLD(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](euclidean-algorithms.md#function-gcld-rngdiffopelt-rngdiffopelt)

    - [`ExtendedGreatestCommonLeftDivisor(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt, RngDiffOpElt`](euclidean-algorithms.md#function-extendedgreatestcommonleftdivisor-rngdiffopelt-rngdiffopelt)

    - [`Example GCRD GCLD`](euclidean-algorithms.md#example-ex-5508d6)

  - [Least Common Left Multiples](euclidean-algorithms.md#least-common-left-multiples)

    - [`LeastCommonLeftMultiple(L): RngDiffOpElt → RngDiffOpElt`](euclidean-algorithms.md#function-leastcommonleftmultiple-rngdiffopelt)

    - [`LeastCommonLeftMultiple(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](euclidean-algorithms.md#function-leastcommonleftmultiple-rngdiffopelt-rngdiffopelt)

    - [`LCLM(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt`](euclidean-algorithms.md#function-lclm-rngdiffopelt-rngdiffopelt)

    - [`ExtendedLeastCommonLeftMultiple(A, B): RngDiffOpElt, RngDiffOpElt → RngDiffOpElt, RngDiffOpElt, RngDiffOpElt`](euclidean-algorithms.md#function-extendedleastcommonleftmultiple-rngdiffopelt-rngdiffopelt)

    - [`ExtendedLeastCommonLeftMultiple(S): [RngDiffOpElt] → RngDiffOpElt, SeqEnum`](euclidean-algorithms.md#function-extendedleastcommonleftmultiple-rngdiffopelt)

    - [`Example LCLM`](euclidean-algorithms.md#example-ex-22d5f6)

    - [`Example LCLM Conjugates`](euclidean-algorithms.md#example-ex-138207)

- [Related Matrices](related-matrix.md)

  - [`CompanionMatrix(L): RngDiffOpElt → AlgMatElt`](related-matrix.md#function-companionmatrix-rngdiffopelt)

  - [`Example Companion Matrix`](related-matrix.md#example-ex-62793f)

- [Singular Places and Indicial Polynomials](singularities.md)

  - [Singular Places](singularities.md#singular-places)

    - [`IsRegularPlace(L, p): RngDiffOpElt, PlcFunElt → BoolElt`](singularities.md#function-isregularplace-rngdiffopelt-plcfunelt)

    - [`IsRegularSingularPlace(L, p): RngDiffOpElt, PlcFunElt → BoolElt`](singularities.md#function-isregularsingularplace-rngdiffopelt-plcfunelt)

    - [`IsIrregularSingularPlace(L, p): RngDiffOpElt, PlcFunElt → BoolElt`](singularities.md#function-isirregularsingularplace-rngdiffopelt-plcfunelt)

    - [`SetsOfSingularPlaces(L): RngDiffOpElt → SetEnum, SetEnum`](singularities.md#function-setsofsingularplaces-rngdiffopelt)

    - [`IsFuchsianOperator(L): RngDiffOpElt → BoolElt, SetEnum`](singularities.md#function-isfuchsianoperator-rngdiffopelt)

    - [`IsRegularSingularOperator(L): RngDiffOpElt → BoolElt, SetEnum`](singularities.md#function-isregularsingularoperator-rngdiffopelt)

    - [`Example Singularities`](singularities.md#example-ex-3dba78)

    - [`Example Regular Singular DLSR`](singularities.md#example-ex-5e64d4)

  - [Indicial Polynomials](singularities.md#indicial-polynomials)

    - [`IndicialPolynomial(L, p): RngDiffOpElt, PlcFunElt → RngElt`](singularities.md#function-indicialpolynomial-rngdiffopelt-plcfunelt)

    - [`Example Indicial Pol`](singularities.md#example-ex-eadf1f)

- [Rational Solutions](rational-sols.md)

  - [`RationalSolutions(L): RngDiffOpElt → SeqEnum`](rational-sols.md#function-rationalsolutions-rngdiffopelt)

  - [`HasRationalSolutions(L, g): RngDiffOpElt, RngElt → BoolElt, RngElt, SeqEnum`](rational-sols.md#function-hasrationalsolutions-rngdiffopelt-rngelt)

  - [`Example Rational Sols`](rational-sols.md#example-ex-a82ef2)

- [Newton Polygons](newton-polygon.md)

  - [`NewtonPolygon(L): RngDiffOpElt → NwtnPgon, RingDiffOpElt`](newton-polygon.md#function-newtonpolygon-rngdiffopelt)

  - [`NewtonPolygon(L, p): RngDiffOpElt, PlcFunElt → NwtnPgon, RingDiffOpElt`](newton-polygon.md#function-newtonpolygon-rngdiffopelt-plcfunelt)

  - [`NewtonPolynomial(F): NwtnPgonFace → RngUPolElt`](newton-polygon.md#function-newtonpolynomial-nwtnpgonface)

  - [`NewtonPolynomials(L): RngDiffOpElt → SeqEnum, SeqEnum`](newton-polygon.md#function-newtonpolynomials-rngdiffopelt)

  - [`Example Newton Pgns 1`](newton-polygon.md#example-ex-893538)

  - [`Example Newton Pgns 2`](newton-polygon.md#example-ex-692056)

  - [`Example Newton Pgns 3`](newton-polygon.md#example-ex-312e21)

- [Symmetric Powers](symmetric-power.md)

  - [`SymmetricPower(L, m): RngDiffOpElt, RngIntElt → RngDiffOpElt`](symmetric-power.md#function-symmetricpower-rngdiffopelt-rngintelt)

  - [`Example Symmetric Power`](symmetric-power.md#example-ex-6160ae)

- [Differential Operators of Algebraic Functions](operators-of-functions.md)

  - [`DifferentialOperator(f): RngUPolElt → RngDiffOpElt`](operators-of-functions.md#function-differentialoperator-rngupolelt)

  - [`Example Diff Op Algebraic Function`](operators-of-functions.md#example-ex-a16c5a)

- [Factorisation of Operators over Differential Laurent Series Rings](operators-factorisation-over-laurent-series-ring.md)

  - [Slope Valuation of an Operator](operators-factorisation-over-laurent-series-ring.md#slope-valuation-of-an-operator)

    - [`SlopeValuation(L,s): RngDiffOpElt, RngElt → FldRatElt`](operators-factorisation-over-laurent-series-ring.md#function-slopevaluation-rngdiffopelt-rngelt)

    - [`Example Diff Op Slope Valuation`](operators-factorisation-over-laurent-series-ring.md#example-ex-402dde)

  - [Coprime Index $1$ and LCLM Factorisation](operators-factorisation-over-laurent-series-ring.md#coprime-index-1-and-lclm-factorisation)

    - [`Factorisation(L): RngDiffOpElt → SeqEnum, SeqEnum`](operators-factorisation-over-laurent-series-ring.md#function-factorisation-rngdiffopelt)

    - [`Factorization(L): RngDiffOpElt → SeqEnum, SeqEnum`](operators-factorisation-over-laurent-series-ring.md#function-factorization-rngdiffopelt)

    - [`Example Diff Op Factorisation LCLM 1`](operators-factorisation-over-laurent-series-ring.md#example-ex-502230)

    - [`Example Diff Op Factorisation LCLM 2`](operators-factorisation-over-laurent-series-ring.md#example-ex-6f1759)

    - [`Example Diff Op Factorisation LCLM 3`](operators-factorisation-over-laurent-series-ring.md#example-ex-6434dc)

    - [`Example Diff Op Factorisation LCLM 4`](operators-factorisation-over-laurent-series-ring.md#example-ex-c4b3ef)

    - [`Example Diff Op Factorisation LCLM 5`](operators-factorisation-over-laurent-series-ring.md#example-ex-8a9dd9)

    - [`Example Diff Op Factorisation LCLM 6`](operators-factorisation-over-laurent-series-ring.md#example-ex-fee1e8)

  - [Right Hand Factors of Operators](operators-factorisation-over-laurent-series-ring.md#right-hand-factors-of-operators)

    - [`RightHandFactors(L): RngDiffOpElt → SeqEnum, [BoolElt]`](operators-factorisation-over-laurent-series-ring.md#function-righthandfactors-rngdiffopelt)

    - [`Example Diff Op Righthandfactors 1`](operators-factorisation-over-laurent-series-ring.md#example-ex-3c08eb)

    - [`Example Diff Op Righthandfactors 2`](operators-factorisation-over-laurent-series-ring.md#example-ex-014248)

    - [`Example Diff Op Righthandfactors 3`](operators-factorisation-over-laurent-series-ring.md#example-ex-69cc8c)

    - [`Example Diff Op Righthandfactors 4`](operators-factorisation-over-laurent-series-ring.md#example-ex-6d4ec1)

    - [`Example Diff Op Righthandfactors 5`](operators-factorisation-over-laurent-series-ring.md#example-ex-987714)

    - [`Example Diff Op Righthandfactors 5`](operators-factorisation-over-laurent-series-ring.md#example-ex-826452)