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diffalg[preparation_polynomial] - compute preparation polynomial

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:

R:=differential_ringderivations=x,y,ranking=u,A:

p:=16ux,yux,x2uy,y2+uy,yux,x+u[]24u[]x2y2

p:=16ux,yux,x2uy,y2ux,x+uy,y+u[]2x2y2+4u[]

(1)

equationsRosenfeld_Groebnerp,R

x2u[]2y2u[]2+4u[]3+16ux,x2ux,y16ux,yuy,y216ux,xux,y+16ux,yuy,y,x2y2+4u[],u[]

(2)

preparation_polynomialp,u[],R

4_A[]316_Ay,y2_Ax,y+16_Ax,y_Ax,x2+x2y2_A[]2+16_Ay,y_Ax,y16_Ax,y_Ax,x

(3)

preparation_polynomialp,A[]=4u[]x2y2,R

A[]34Ay,y2Ax,y+4Ax,yAx,x2+2x2+2y2A[]2+x4+2x2y2+y4A[]

(4)

Studying the degree in A (or _A) and its derivatives in these preparation polynomials, we can deduce that ux,y=14x2+14y2 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.

R:=differential_ringranking=y,A,derivations=x:

p:=3yx4yx$2yx$424yx4yx$32yx$4+6yx3yx$22yx$3yx$4+24yx2yx$24yx$412yx3yx$2yx$3329yx2yx$23yx$32+12yx$27

p:=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

(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)

q:=3yx,x4+yx2yx,x,x2

q:=yx2yx,x,x2+3yx,x4

(7)

preparation_polynomialp,A[]=q,R

32yxyx,xyx,x,x32yx,x3A[]28yx2yx,x,xA[]Ax+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

12A[]7+24A[]4Ax,xyx229A[]3Ax2yx2+6A[]2AxAx,xyx312A[]Ax3yx3+3A[]Ax,x2yx44Ax2Ax,xyx4

(10)

preparation_polynomialq,A[]=yx,x,R

3A[]4+Ax2yx2

(11)

See Also

diffalg(deprecated), diffalg(deprecated)/differential_algebra, diffalg(deprecated)/differential_ring, diffalg(deprecated)/essential_components, diffalg(deprecated)/Rosenfeld_Groebner, DifferentialAlgebra[PreparationEquation]


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