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diffalg

  

preparation_polynomial

  

compute preparation polynomial

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

preparation_polynomial (p, a, R, 'm' )

preparation_polynomial (p, A=a, R, 'm' )

Parameters

p

-

differential polynomial in R

a

-

regular differential polynomial in R

R

-

differential polynomial ring

m

-

(optional) name

A

-

derivative of order zero in R

Description

• 

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

• 

The function preparation_polynomial computes a preparation polynomial of p with respect to a.

• 

The preparation polynomial of p with respect to a is a sort of expansion of p according to mparama and its derivatives. It plays a prominent role in the determination of the essential components of the radical differential ideal generated by a single differential polynomial.

• 

A differential polynomial a is said to be regular if it has no common factor with its separant. This property is therefore dependent on the ranking defined on R.

• 

If A is omitted, the preparation polynomial appears with an  indeterminate  (local variable) looking like  _A.

• 

If A is  specified, the preparation polynomial is in the  differential indeterminate A. Then, A, nor its derivatives, should appear in p nor a.

• 

Assume that preparation_polynomial(p, a, R, 'm') = c1M1_A+....+ckMk_A, where the Mi are differential monomials in _A and the ci are polynomials in R. Then

  

- mp=c1M1a+....+ckMka, where m belongs to R.

  

- The ci are not reduced to zero by a, and therefore do not belong to the general component of a.

  

- m is a power product of factors of the initial and separant of a).

• 

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

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

The preparation polynomial is used to determine the essential singular zeros of a differential polynomial.

withdiffalg:

Rdifferential_ringderivations=x,y,ranking=u,A:

p16ux,yux,x2uy,y2+uy,yux,x+u24ux2y2

p16ux,yux,x2uy,y2ux,x+uy,y+u2x2y2+4u

(1)

equationsRosenfeld_Groebnerp,R

x2u2y2u2+4u3+16ux,x2ux,y16ux,yuy,y216ux,xux,y+16ux,yuy,y,x2y2+4u,u

(2)

preparation_polynomialp,u,R

4_A316_Ay,y2_Ax,y+16_Ax,y_Ax,x2+x2y2_A2+16_Ay,y_Ax,y16_Ax,y_Ax,x

(3)

preparation_polynomialp,A=4ux2y2,R

A34Ay,y2Ax,y+4Ax,yAx,x2+2x2+2y2A2+x4+2x2y2+y4A

(4)

Studying the degree in A (or _A) and its derivatives in these preparation polynomials, we can deduce that ux,y=x24+y24 is an essential singular zero of p while ux,y=0 is not.

The preparation polynomial can be used to further study the relationships between the general zero and the singular zeros of p.

Rdifferential_ringranking=y,A,derivations=x:

p3yx4y`$`x,2y`$`x,424yx4y`$`x,32y`$`x,4+6yx3y`$`x,22y`$`x,3y`$`x,4+24yx2y`$`x,24y`$`x,412yx3y`$`x,2y`$`x,3329yx2y`$`x,23y`$`x,32+12y`$`x,27

p3yx4yx,xyx,x,x,x24yx4yx,x,x2yx,x,x,x+6yx3yx,x2yx,x,xyx,x,x,x12yx3yx,xyx,x,x3+24yx2yx,x4yx,x,x,x29yx2yx,x3yx,x,x2+12yx,x7

(5)

equationsRosenfeld_Groebnerp,R

3yx4yx,xyx,x,x,x24yx4yx,x,x2yx,x,x,x+6yx3yx,x2yx,x,xyx,x,x,x12yx3yx,xyx,x,x3+24yx2yx,x4yx,x,x,x29yx2yx,x3yx,x,x2+12yx,x7,yx2yx,x,x2+3yx,x4,yx,x

(6)

q3yx,x4+yx2yx,x,x2

qyx2yx,x,x2+3yx,x4

(7)

preparation_polynomialp,A=q,R

32yx,xyx,x,xyx32yx,x3A28yx2yx,x,xAAx+3yx,xyx2Ax2+96yxyx,x5yx,x,x+96yx,x7A

(8)

equationsessential_componentsp,R

3yx4yx,xyx,x,x,x24yx4yx,x,x2yx,x,x,x+6yx3yx,x2yx,x,xyx,x,x,x12yx3yx,xyx,x,x3+24yx2yx,x4yx,x,x,x29yx2yx,x3yx,x,x2+12yx,x7,yx2yx,x,x2+3yx,x4

(9)

The general zero of q is an essential singular zero of p while the general zero of yx,x is not. Thus, the straight lines yx=_C1x+_C2, zeros of yx,x, must be limits of either some non singular zeros of p or of the non singular zeros of q. Again studying the degrees of the preparation polynomials of p and q we can deduce that the straight lines are in fact limits of the non singular zeros of both (cf. [Kolchin]).

preparation_polynomialp,A=yx,x,R

12A7+24yx2A4Ax,x29yx2A3Ax2+6yx3A2AxAx,x12yx3AAx3+3yx4AAx,x24yx4Ax2Ax,x

(10)

preparation_polynomialq,A=yx,x,R

3A4+yx2Ax2

(11)

See Also

diffalg(deprecated)

diffalg(deprecated)/differential_algebra

diffalg(deprecated)/differential_ring

diffalg(deprecated)/essential_components

diffalg(deprecated)/Rosenfeld_Groebner

DifferentialAlgebra[PreparationEquation]