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') = $\mathrm{c1}\mathrm{M1}\left(\mathrm{\_A}\right)+\mathrm{....}+\mathrm{ck}\mathrm{Mk}\left(\mathrm{\_A}\right)$, where the Mi are differential monomials in $\mathrm{\_A}$ and the $\mathrm{ci}$ are polynomials in R. Then

- $mp=\mathrm{c1}\mathrm{M1}\left(a\right)+\mathrm{....}+\mathrm{ck}\mathrm{Mk}\left(a\right)$, where m belongs to R.

- The $\mathrm{ci}$ 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.

$\mathrm{with}\left(\mathrm{diffalg}\right)\:$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[u\,A\right]\right)\:$

$p≔16u\left[x,y\right]\left({u\left[x,x\right]}^2-{u\left[y,y\right]}^2+u\left[y,y\right]-u\left[x,x\right]\right)+{u\left[\right]}^2\left(4u\left[\right]-x^2-y^2\right)$

$p≔16u_{x,y}\left(u_{x,x}^2-u_{y,y}^2-u_{x,x}+u_{y,y}\right)+{u\left[\right]}^2\left(-x^2-y^2+4u\left[\right]\right)$

$\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[p\right]\,R\right)\right)$

$\left[\left[-x^2{u\left[\right]}^2-y^2{u\left[\right]}^2+4{u\left[\right]}^3+16u_{x,x}^2{u}_{x,y}-16u_{x,y}{u}_{y,y}^2-16u_{x,x}{u}_{x,y}+16u_{x,y}{u}_{y,y}\right]\,\left[-x^2-y^2+4u\left[\right]\right]\,\left[u\left[\right]\right]\right]$

$\mathrm{preparation\_polynomial}\left(p\,u\left[\right]\,R\right)$

$4{\mathrm{\_A}\left[\right]}^3-16{\mathrm{\_A}}_{y,y}^2{\mathrm{\_A}}_{x,y}+16{\mathrm{\_A}}_{x,y}{\mathrm{\_A}}_{x,x}^2+\left(-x^2-y^2\right){\mathrm{\_A}\left[\right]}^2+16{\mathrm{\_A}}_{y,y}{\mathrm{\_A}}_{x,y}-16{\mathrm{\_A}}_{x,y}{\mathrm{\_A}}_{x,x}$

$\mathrm{preparation\_polynomial}\left(p\,A\left[\right]=4u\left[\right]-x^2-y^2\,R\right)$

${A\left[\right]}^3-4A_{y,y}^2{A}_{x,y}+4A_{x,y}{A}_{x,x}^2+\left(2x^2+2y^2\right){A\left[\right]}^2+\left(x^4+2x^2{y}^2+y^4\right)A\left[\right]$

$R≔\mathrm{differential\_ring}\left(\mathrm{ranking}=\left[y\,A\right]\,\mathrm{derivations}=\left[x\right]\right)\:$

$p≔3{y\left[x\right]}^{4}y\left[\mathrm{`\$`}\left(x\,2\right)\right]{y\left[\mathrm{`\$`}\left(x\,4\right)\right]}^2-4{y\left[x\right]}^4{y\left[\mathrm{`\$`}\left(x\,3\right)\right]}^{2}y\left[\mathrm{`\$`}\left(x\,4\right)\right]+6{y\left[x\right]}^3{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^{2}y\left[\mathrm{`\$`}\left(x\,3\right)\right]y\left[\mathrm{`\$`}\left(x\,4\right)\right]+24{y\left[x\right]}^2{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^{4}y\left[\mathrm{`\$`}\left(x\,4\right)\right]-12{y\left[x\right]}^{3}y\left[\mathrm{`\$`}\left(x\,2\right)\right]{y\left[\mathrm{`\$`}\left(x\,3\right)\right]}^3-29{y\left[x\right]}^2{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^3{y\left[\mathrm{`\$`}\left(x\,3\right)\right]}^2+12{y\left[\mathrm{`\$`}\left(x\,2\right)\right]}^7$

$p≔3y_x^4{y}_{x,x}{y}_{x,x,x,x}^2-4y_x^4{y}_{x,x,x}^2{y}_{x,x,x,x}+6y_x^3{y}_{x,x}^2{y}_{x,x,x}{y}_{x,x,x,x}-12y_x^3{y}_{x,x}{y}_{x,x,x}^3+24y_x^2{y}_{x,x}^4{y}_{x,x,x,x}-29y_x^2{y}_{x,x}^3{y}_{x,x,x}^2+12y_{x,x}^7$

$\mathrm{equations}\left(\mathrm{Rosenfeld\_Groebner}\left(\left[p\right]\,R\right)\right)$

$\left[\left[3y_x^4{y}_{x,x}{y}_{x,x,x,x}^2-4y_x^4{y}_{x,x,x}^2{y}_{x,x,x,x}+6y_x^3{y}_{x,x}^2{y}_{x,x,x}{y}_{x,x,x,x}-12y_x^3{y}_{x,x}{y}_{x,x,x}^3+24y_x^2{y}_{x,x}^4{y}_{x,x,x,x}-29y_x^2{y}_{x,x}^3{y}_{x,x,x}^2+12y_{x,x}^7\right]\,\left[y_x^2{y}_{x,x,x}^2+3y_{x,x}^4\right]\,\left[y_{x,x}\right]\right]$

$q≔3{y\left[x,x\right]}^4+{y\left[x\right]}^2{y\left[x,x,x\right]}^2$

$q≔y_x^2{y}_{x,x,x}^2+3y_{x,x}^4$

$\mathrm{preparation\_polynomial}\left(p\,A\left[\right]=q\,R\right)$

$\left(-32y_{x,x}{y}_{x,x,x}{y}_x-32y_{x,x}^3\right){A\left[\right]}^2-8y_x^2{y}_{x,x,x}A\left[\right]A_x+3y_{x,x}{y}_x^2{A}_x^2+\left(96y_x{y}_{x,x}^5{y}_{x,x,x}+96y_{x,x}^7\right)A\left[\right]$

$\mathrm{equations}\left(\mathrm{essential\_components}\left(p\,R\right)\right)$

$\left[\left[3y_x^4{y}_{x,x}{y}_{x,x,x,x}^2-4y_x^4{y}_{x,x,x}^2{y}_{x,x,x,x}+6y_x^3{y}_{x,x}^2{y}_{x,x,x}{y}_{x,x,x,x}-12y_x^3{y}_{x,x}{y}_{x,x,x}^3+24y_x^2{y}_{x,x}^4{y}_{x,x,x,x}-29y_x^2{y}_{x,x}^3{y}_{x,x,x}^2+12y_{x,x}^7\right]\,\left[y_x^2{y}_{x,x,x}^2+3y_{x,x}^4\right]\right]$

$\mathrm{preparation\_polynomial}\left(p\,A\left[\right]=y\left[x,x\right]\,R\right)$

$12{A\left[\right]}^7+24y_x^2{A\left[\right]}^4{A}_{x,x}-29y_x^2{A\left[\right]}^3{A}_x^2+6y_x^3{A\left[\right]}^2{A}_x{A}_{x,x}-12y_x^{3}A\left[\right]A_x^3+3y_x^{4}A\left[\right]A_{x,x}^2-4y_x^4{A}_x^2{A}_{x,x}$

$\mathrm{preparation\_polynomial}\left(q\,A\left[\right]=y\left[x,x\right]\,R\right)$

$3{A\left[\right]}^4+y_x^2{A}_x^2$