compute a characteristic decomposition of the radical differential ideal generated by a finite set of differential polynomials

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diffalg[Rosenfeld_Groebner] - compute a characteristic decomposition of the radical differential ideal generated by a finite set of differential polynomials

	Calling Sequence
	Rosenfeld_Groebner (S, H, R, J)

Parameters

S	-	list or set of differential polynomials of R
H	-	(optional) list or a set of differential polynomials of R
R	-	differential polynomial ring
J	-	(optional) radical differential ideal

Description

•	For an informal presentation, see the diffalg overview.

•	Rosenfeld_Groebner computes a characteristic decomposition of the radical differential ideal P = {S}:(H)^infinity.

•	If the parameter H is omitted, Rosenfeld_Groebner computes a characteristic decomposition of the radical differential ideal P={S} generated by the differential polynomials of S.

•	R is a differential polynomial ring constructed with the differential_ring command.

•	The output of Rosenfeld_Groebner depends on the ranking defined on R.

•	Rosenfeld_Groebner returns a list of characterizable differential ideals.

The empty list denotes the unit ideal (meaning that there is no solution).

Each characterizable differential ideal is stored in a table. Only the name of the table () is printed on the screen. To access their defining characteristic sets you can use the commands rewrite_rules, equations, and inequations.

•	If the fourth parameter J is present, it is assumed to be another representation of with respect to another ranking. It is used to spare some splittings. It can be used to speed up the computation, for example, if there is a natural ranking to compute the representation of .

•	The command with(diffalg,Rosenfeld_Groebner) allows the use of the abbreviated form of this command.

Examples

The first example illustrates how the Rosenfeld_Groebner command splits a system of differential equations into a system representing the general solution and systems representing the singular solutions.

(1)

(2)

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(4)

[-y(x)+x*(diff(y(x), x))+(diff(y(x), x))^2+diff(z(x), x), (diff(y(x), x))^3+2*(diff(y(x), x))^2*x-(diff(y(x), x))*y(x)+(diff(y(x), x))*x^2+z(x)-y(x)*x], [3*(diff(y(x), x))^2+4*x*(diff(y(x), x))-y(x)+x^2]

(5)

[6*(diff(y(x), x))*y(x)+2*(diff(y(x), x))*x^2-9*z(x)+7*y(x)*x+2*x^3, 27*z(x)^2-18*z(x)*y(x)*x-4*z(x)*x^3-4*y(x)^3-y(x)^2*x^2], [3*y(x)+x^2, 27*z(x)-9*y(x)*x-2*x^3]

(6)

(7)

To obtain the characterizable differential ideal representing the general solution alone, we can proceed as follows.

(8)

It is sometimes the case that the radical differential ideal generated by S is prime. This can be proved by exhibiting a ranking for which the characteristic decomposition of P consists of only one orthonomic characterizable differential ideal.

Before computing a representation of P with respect to the ranking of R, it may be useful to proceed as follows. Search for a ranking for which the characteristic decomposition is as described above. Assign J this computed characteristic decomposition. Then call Rosenfeld_Groebner with J as fourth parameter.

With such a fourth parameter, whatever the ranking of R is, the computed representation of P consists of only one characterizable differential ideal.

If J consists of a single non-orthonomic component or has more than one characterizable component, Rosenfeld_Groebner uses the information to avoid unnecessary splittings.

The example below illustrates this behavior for Euler's equations for an incompressible fluid in two dimensions.

(9)

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(11)

(12)

(13)

(14)

	See Also
	diffalg(deprecated), diffalg(deprecated)/differential_algebra, diffalg(deprecated)/Rosenfeld_Groebner_options, diffalg(deprecated)[differential_ring], diffalg(deprecated)[equations], diffalg(deprecated)[inequations], diffalg(deprecated)[is_orthonomic], diffalg(deprecated)[rewrite_rules],