## `detools::cartan` -- Cartan characters of a differential equation

#### Introduction

`detools::cartan` determines the Cartan characters of a differential equation either from the indices of the symbol or from the Hilbert polynomial.

#### Call(s)

`detools::cartan(n, m, q, beta)`
`detools::cartan(n, hp)`

#### Parameters

`n` | - | number of independent variables: a positive integer.
`m` | - | number of dependent variables: a positive integer.
`q` | - | order of the equation: a positive integer.
`beta` | - | indices of symbol: a list of nonnegative integer.
`hp` | - | Hilbert polynomial: a univariate polynomial (type `DOM_POLY`) with rational coefficients.

#### Returns

a list of nonnegative integers.

#### Related Functions

`detools::arbFuns`, `detools::hilbert`

#### Details

- `detools::cartan` determines the Cartan characters of a differential equation from either the indices of the symbol or the Hilbert polynomial. In the first case, the number of independent and dependent variables, respectively, of the equation and its order must be given.
- In the second form to call `detools::cartan` the number of independent variables suffices, as `detools::cartan` assumes that the Hilbert polynomial is given relative to the order of the differential equation, i.e. as a function of q+r.

#### Example 1

The indices of the symbol of Maxwell's equations of electrodynamics (a first order system in four independent and six dependent variables) are `[0,0,2,6]`. They can be converted into Cartan characters with the following command.

    >> detools::cartan(4, 6, 1, [0, 0, 2, 6])

                                   [6, 6, 4, 0]

Alternatively, we may start with the Hilbert polynomial of Maxwell's equations: H(1+r)=2r^2+12r+16.

    >> detools::cartan(4, poly(2*r^2 + 12*r + 16, [r], Dom::Rational))

                                   [6, 6, 4, 0]

#### Background

- Cartan characters are used to measure the size of the (formal) solution space of a system of differential equations. They appear in the so-called formal theory of differential equations and are based on an analysis of power series solutions. More details can be found in the article:
  - W.M. Seiler: On the arbitrariness of the general solution of an involutive partial differential equation, Journal of Mathematical Physics 35 (1994) 486-498

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