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
qand with characters
- alphaHilbert: SparseUnivariatePolynomial Fraction Integer -> List NonNegativeInteger
alphaHilbert(hp)computes the Cartan characters for the Hilbert polynomial
- arbFunctions: (NonNegativeInteger, Integer, List NonNegativeInteger) -> List Integer
arbFunctions(q, j, cc)uses the Cartan characters
ccto compute the number of arbitrary functions of differentiation order
jin the general solution of a differential equation of order
- 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
mdependent variables involutive.
- 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
ISysfor the involutive system without prolongations, its order
Order, the number
NumProjof needed projections and the Cartan characters
- complete: JetDifferentialEquation(JB, D) -> Void
deto an involutive equation. No result is returned; the display depends of the setting of the output flags with setOutput.
- gauge: (NonNegativeInteger, Integer, List NonNegativeInteger) -> List Integer
gauge(q, j, gamma)computes the gauge corrections to the number of arbitrary functions of differentiation order
jfor a system of order
- gaugeHilbert: (NonNegativeInteger, List NonNegativeInteger) -> SparseUnivariatePolynomial Fraction Integer
gaugeHilbert(q, gamma)computes the gauge correction to the Hilbert polynomial for a system of order
- hilbert: List NonNegativeInteger -> SparseUnivariatePolynomial Fraction Integer
hilbert(cc)computes the Hilbert polynomial to the Cartan characters
- 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 > 10then 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.