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diffalg[preparation_polynomial] - compute preparation polynomial
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Calling Sequence
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preparation_polynomial (p, a, R, 'm' )
preparation_polynomial (p, A=a, R, 'm' )
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Parameters
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p
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differential polynomial in R
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a
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regular differential polynomial in R
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R
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differential polynomial ring
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m
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(optional) name
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A
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derivative of order zero in R
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Description
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The function preparation_polynomial computes a preparation polynomial of p with respect to a.
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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.
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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.
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If A is omitted, the preparation polynomial appears with an indeterminate (local variable) looking like _A.
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If A is specified, the preparation polynomial is in the differential indeterminate A. Then, A, nor its derivatives, should appear in p nor a.
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-  , where m belongs to R.
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- The  are not reduced to zero by a, and therefore do not belong to the general component of a.
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- m is a power product of factors of the initial and separant of a).
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The command with(diffalg,preparation_polynomial) allows the use of the abbreviated form of this command.
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Examples
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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.
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![R := differential_ring(derivations = [x, y], ranking = [u, A])](/support/helpjp/helpview.aspx?si=7394/file07266/math209.png)
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![p := 16*u[x, y]*(u[x, x]^2-u[y, y]^2+u[y, y]-u[x, x])+u[]^2*(4*u[]-x^2-y^2)](/support/helpjp/helpview.aspx?si=7394/file07266/math213.png)
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![p := 16*u[x, y]*(u[x, x]^2-u[y, y]^2+u[y, y]-u[x, x])+u[]^2*(4*u[]-x^2-y^2)](/support/helpjp/helpview.aspx?si=7394/file07266/math216.png)
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![equations(Rosenfeld_Groebner([p], R))](/support/helpjp/helpview.aspx?si=7394/file07266/math220.png)
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![[[16*u[x, y]*u[x, x]^2-16*u[x, y]*u[y, y]^2+16*u[x, y]*u[y, y]-16*u[x, y]*u[x, x]+4*u[]^3-u[]^2*x^2-u[]^2*y^2], [4*u[]-x^2-y^2], [u[]]]](/support/helpjp/helpview.aspx?si=7394/file07266/math223.png)
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![preparation_polynomial(p, u[], R)](/support/helpjp/helpview.aspx?si=7394/file07266/math227.png)
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![4*_A[]^3-16*_A[y, y]^2*_A[x, y]+16*_A[x, y]*_A[x, x]^2+(-x^2-y^2)*_A[]^2+16*_A[y, y]*_A[x, y]-16*_A[x, y]*_A[x, x]](/support/helpjp/helpview.aspx?si=7394/file07266/math230.png)
| (3) |
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![preparation_polynomial(p, A[] = 4*u[]-x^2-y^2, R)](/support/helpjp/helpview.aspx?si=7394/file07266/math234.png)
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![A[]^3-4*A[y, y]^2*A[x, y]+4*A[x, y]*A[x, x]^2+(2*y^2+2*x^2)*A[]^2+(x^4+y^4+2*x^2*y^2)*A[]](/support/helpjp/helpview.aspx?si=7394/file07266/math237.png)
| (4) |
Studying the degree in  (or  ) and its derivatives in these preparation polynomials, we can deduce that  is an essential singular zero of  while  is not.
The preparation polynomial can be used to further study the relationships between the general zero and the singular zeros of  .
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![R := differential_ring(ranking = [y, A], derivations = [x])](/support/helpjp/helpview.aspx?si=7394/file07266/math257.png)
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![p := 3*y[x]^4*y[`$`(x, 2)]*y[`$`(x, 4)]^2-4*y[x]^4*y[`$`(x, 3)]^2*y[`$`(x, 4)]+6*y[x]^3*y[`$`(x, 2)]^2*y[`$`(x, 3)]*y[`$`(x, 4)]+24*y[x]^2*y[`$`(x, 2)]^4*y[`$`(x, 4)]-12*y[x]^3*y[`$`(x, 2)]*y[`$`(x, 3)]^3-29*y[x]^2*y[`$`(x, 2)]^3*y[`$`(x, 3)]^2+12*y[`$`(x, 2)]^7](/support/helpjp/helpview.aspx?si=7394/file07266/math261.png)
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![p := 3*y[x]^4*y[x, x]*y[x, x, x, x]^2-4*y[x]^4*y[x, x, x]^2*y[x, x, x, x]+6*y[x]^3*y[x, x]^2*y[x, x, x]*y[x, x, x, x]+24*y[x]^2*y[x, x]^4*y[x, x, x, x]-12*y[x]^3*y[x, x]*y[x, x, x]^3-29*y[x]^2*y[x, x]^3*y[x, x, x]^2+12*y[x, x]^7](/support/helpjp/helpview.aspx?si=7394/file07266/math264.png)
| (5) |
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![equations(Rosenfeld_Groebner([p], R))](/support/helpjp/helpview.aspx?si=7394/file07266/math268.png)
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![[[3*y[x]^4*y[x, x]*y[x, x, x, x]^2-4*y[x]^4*y[x, x, x]^2*y[x, x, x, x]+6*y[x]^3*y[x, x]^2*y[x, x, x]*y[x, x, x, x]+24*y[x]^2*y[x, x]^4*y[x, x, x, x]-12*y[x]^3*y[x, x]*y[x, x, x]^3-29*y[x]^2*y[x, x]^3*y[x, x, x]^2+12*y[x, x]^7], [y[x, x, x]^2*y[x]^2+3*y[x, x]^4], [y[x, x]]]](/support/helpjp/helpview.aspx?si=7394/file07266/math271.png)
| (6) |
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![q := 3*y[x, x]^4+y[x]^2*y[x, x, x]^2](/support/helpjp/helpview.aspx?si=7394/file07266/math275.png)
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![q := y[x, x, x]^2*y[x]^2+3*y[x, x]^4](/support/helpjp/helpview.aspx?si=7394/file07266/math278.png)
| (7) |
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![preparation_polynomial(p, A[] = q, R)](/support/helpjp/helpview.aspx?si=7394/file07266/math282.png)
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![(-32*y[x, x]^3-32*y[x, x]*y[x, x, x]*y[x])*A[]^2-8*y[x]^2*y[x, x, x]*A[]*A[x]+3*y[x, x]*y[x]^2*A[x]^2+(96*y[x]*y[x, x, x]*y[x, x]^5+96*y[x, x]^7)*A[]](/support/helpjp/helpview.aspx?si=7394/file07266/math285.png)
| (8) |
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![[[3*y[x]^4*y[x, x]*y[x, x, x, x]^2-4*y[x]^4*y[x, x, x]^2*y[x, x, x, x]+6*y[x]^3*y[x, x]^2*y[x, x, x]*y[x, x, x, x]+24*y[x]^2*y[x, x]^4*y[x, x, x, x]-12*y[x]^3*y[x, x]*y[x, x, x]^3-29*y[x]^2*y[x, x]^3*y[x, x, x]^2+12*y[x, x]^7], [y[x, x, x]^2*y[x]^2+3*y[x, x]^4]]](/support/helpjp/helpview.aspx?si=7394/file07266/math292.png)
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The general zero of  is an essential singular zero of  while the general zero of ![y[x, x]](/support/helpjp/helpview.aspx?si=7394/file07266/math299.png) is not. Thus, the straight lines  , zeros of ![y[x, x]](/support/helpjp/helpview.aspx?si=7394/file07266/math303.png) , must be limits of either some non singular zeros of  or of the non singular zeros of  . Again studying the degrees of the preparation polynomials of  and  we can deduce that the straight lines are in fact limits of the non singular zeros of both (cf. [Kolchin]).
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![preparation_polynomial(p, A[] = y[x, x], R)](/support/helpjp/helpview.aspx?si=7394/file07266/math319.png)
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![12*A[]^7+24*y[x]^2*A[]^4*A[x, x]-29*y[x]^2*A[]^3*A[x]^2+6*y[x]^3*A[]^2*A[x]*A[x, x]-12*y[x]^3*A[]*A[x]^3+3*y[x]^4*A[]*A[x, x]^2-4*y[x]^4*A[x]^2*A[x, x]](/support/helpjp/helpview.aspx?si=7394/file07266/math322.png)
| (10) |
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![preparation_polynomial(q, A[] = y[x, x], R)](/support/helpjp/helpview.aspx?si=7394/file07266/math326.png)
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![3*A[]^4+y[x]^2*A[x]^2](/support/helpjp/helpview.aspx?si=7394/file07266/math329.png)
| (11) |
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