compute preparation polynomial
preparation_polynomial (p, a, R, 'm' )
preparation_polynomial (p, A=a, R, 'm' )
differential polynomial in R
regular differential polynomial in R
differential polynomial ring
derivative of order zero in R
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') = c1⁢M1⁡_A+....+ck⁢Mk⁡_A, where the Mi are differential monomials in _A and the ci are polynomials in R. Then
- m⁢p=c1⁢M1⁡a+....+ck⁢Mk⁡a, 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.
The preparation polynomial is used to determine the essential singular zeros of a differential polynomial.
R ≔ differential_ring⁡derivations=x,y,ranking=u,A:
p ≔ 16⁢ux,y⁢ux,x2−uy,y2+uy,y−ux,x+u2⁢4⁢u−x2−y2
Studying the degree in A (or _A) and its derivatives in these preparation polynomials, we can deduce that u⁡x,y=x24+y24 is an essential singular zero of p while u⁡x,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_ring⁡ranking=y,A,derivations=x:
p ≔ 3⁢yx4⁢yx$2⁢yx$42−4⁢yx4⁢yx$32⁢yx$4+6⁢yx3⁢yx$22⁢yx$3⁢yx$4+24⁢yx2⁢yx$24⁢yx$4−12⁢yx3⁢yx$2⁢yx$33−29⁢yx2⁢yx$23⁢yx$32+12⁢yx$27
q ≔ 3⁢yx,x4+yx2⁢yx,x,x2
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 y⁡x=_C1⁢x+_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]).
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