CartanKuranishi(JB, D)

JB : JetBundleCategory
D : JetBundleFunctionCategory(JB)

CartanKuranishi is a package for the completion of a given differential equation to an involutive equation. Procedures for Cartan characters and Hilbert polynomial are also provided. Based on the Cartan-Kuranishi theorem as it is used in formal theory.

alpha : (NonNegativeInteger, List(NonNegativeInteger)) -> List(NonNegativeInteger): alpha(q, beta) computes the Cartan characters for a differential equation of order q and with characters beta.

alphaHilbert : SparseUnivariatePolynomial(Fraction(Integer)) -> List(NonNegativeInteger): alphaHilbert(hp) computes the Cartan characters for the Hilbert polynomial hp.

arbFunctions : (NonNegativeInteger, Integer, List(NonNegativeInteger)) -> List(Integer): arbFunctions(q, j, cc) uses the Cartan characters cc to compute the number of arbitrary functions of differentiation order j in the general solution of a differential equation of order q.

bound : (NonNegativeInteger, NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger: bound(n, m, q) computes an upper bound for the number of prolongations needed to make the symbol of an equation of order q with n independent and m dependent variables involutive.

complete : JetDifferentialEquation(JB, D) -> Void: complete(de) completes de to an involutive equation. No result is returned; the display depends of the setting of the output flags with setOutput.

complete2 : JetDifferentialEquation(JB, D) -> Record(IDe : JetDifferentialEquation(JB, D), ISys : List(D), Order : NonNegativeInteger, NumProj : NonNegativeInteger, Dim : NonNegativeInteger, CarChar : List(NonNegativeInteger)): complete2(de) is like complete but returns the involutive equation IDe, a basis ISys for the involutive system without prolongations, its order Order, the number NumProj of needed projections and the Cartan characters CarChar.

gauge : (NonNegativeInteger, Integer, List(NonNegativeInteger)) -> List(Integer): gauge(q, j, gamma) computes the gauge corrections to the number of arbitrary functions of differentiation order j for a system of order q with gamma gauge functions.

gaugeHilbert : (NonNegativeInteger, List(NonNegativeInteger)) -> SparseUnivariatePolynomial(Fraction(Integer)): gaugeHilbert(q, gamma) computes the gauge correction to the Hilbert polynomial for a system of order q with gamma gauge functions.

hilbert : List(NonNegativeInteger) -> SparseUnivariatePolynomial(Fraction(Integer)): hilbert(cc) computes the Hilbert polynomial to the Cartan characters cc.

setOutMode : NonNegativeInteger -> NonNegativeInteger: setOutput(i) controls amount of generated output during the completion algorithm: i = 0 --> no display, i = 1 --> result is displayed, i = 2 --> Cartan characters are displayed, i = 3 --> integrability conditions are traced, i = 4 --> intermediate dimensions are traced, i = 5 --> all intermediate systems are traced, i = 6 --> all intermediate systems and symbols are traced, if i > 10 then TeX output is produced. Default is 0. The old value is returned.

setRedMode : NonNegativeInteger -> NonNegativeInteger: setRedMode(i) sets the flag for the reduction mode. Returns old value. Current values are: i = 0 --> No reduction of integrability conditions etc. i = 1 --> Autoreduction of complete system and reduction of all integrability conditions. Default is 0.

setSimpMode : NonNegativeInteger -> NonNegativeInteger: setSimpMode(i) sets the simplification mode used in JetDifferentialEquation. Returns old value.

stirling : (NonNegativeInteger, NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger: stirling(i, k, q) computes the corresponding modified Stirling number.