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

For an informal presentation, see the diffalg overview.

Rosenfeld_Groebner computes a characteristic decomposition of the radical differential ideal P = {S}:(H)^infinity.

If the parameter H is omitted, Rosenfeld_Groebner computes a characteristic decomposition of the radical differential ideal P={S} generated by the differential polynomials of S.

R is a differential polynomial ring constructed with the differential_ring command.

The output of Rosenfeld_Groebner depends on the ranking defined on R.

Rosenfeld_Groebner returns a list of characterizable differential ideals.

The empty list denotes the unit ideal (meaning that there is no solution).

Each characterizable differential ideal is stored in a table. Only the name of the table ($\mathrm{characterizable}$) is printed on the screen. To access their defining characteristic sets you can use the commands rewrite_rules, equations, and inequations.

If the fourth parameter J is present, it is assumed to be another representation of $P$ with respect to another ranking. It is used to spare some splittings. It can be used to speed up the computation, for example, if there is a natural ranking to compute the representation of $P$.

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

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

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

$R≔\mathrm{ODE\_ring}$

$\mathrm{eq1}≔-y\left(x\right)+x\mathrm{diff}\left(y\left(x\right)\,x\right)+{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2+\mathrm{diff}\left(z\left(x\right)\,x\right)$

$\mathrm{eq1}≔-y\left(x\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2+\frac{\ⅆ}{\ⅆx}z\left(x\right)$

$\mathrm{eq2}≔-z\left(x\right)+x\mathrm{diff}\left(z\left(x\right)\,x\right)+\mathrm{diff}\left(y\left(x\right)\,x\right)\mathrm{diff}\left(z\left(x\right)\,x\right)$

$\mathrm{eq2}≔-z\left(x\right)+x\left(\frac{\ⅆ}{\ⅆx}z\left(x\right)\right)+\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}z\left(x\right)\right)$

$P≔\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{eq1}\,\mathrm{eq2}\right]\,R\right)$

$P≔\left[\mathrm{characterizable}\,\mathrm{characterizable}\,\mathrm{characterizable}\right]$

$\mathrm{equations}\left(P\left[1\right]\right),\mathrm{inequations}\left(P\left[1\right]\right)$

$\left[-y\left(x\right)+x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2+\frac{\ⅆ}{\ⅆx}z\left(x\right)\,{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^3+2{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^{2}x-\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)+\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x^2+z\left(x\right)-y\left(x\right)x\right],\left[x^2+4x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+3{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2-y\left(x\right)\right]$

$\mathrm{equations}\left(P\left[2\right]\right),\mathrm{inequations}\left(P\left[2\right]\right)$

$\left[6\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)+2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x^2-9z\left(x\right)+7y\left(x\right)x+2x^3\,27{z\left(x\right)}^2-18z\left(x\right)y\left(x\right)x-4z\left(x\right)x^3-4{y\left(x\right)}^3-{y\left(x\right)}^2{x}^2\right],\left[3y\left(x\right)+x^2\,-2x^3-9y\left(x\right)x+27z\left(x\right)\right]$

$\mathrm{equations}\left(P\left[3\right]\right),\mathrm{inequations}\left(P\left[3\right]\right)$

$\left[27z\left(x\right)+x^3\,3y\left(x\right)+x^2\right],\left[\right]$

$G≔\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{eq1}\,\mathrm{eq2}\right]\,\left[x^2+4x\mathrm{diff}\left(y\left(x\right)\,x\right)-y\left(x\right)+3{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2\right]\,R\right)$

$G≔\left[\mathrm{characterizable}\right]$

$\mathrm{p1}≔\mathrm{v1}\left[t\right]+\mathrm{v1}\left[\right]\mathrm{v1}\left[x\right]+\mathrm{v2}\left[\right]\mathrm{v1}\left[y\right]+p\left[x\right]\:$

$\mathrm{p2}≔\mathrm{v2}\left[t\right]+\mathrm{v1}\left[\right]\mathrm{v2}\left[x\right]+\mathrm{v2}\left[\right]\mathrm{v2}\left[y\right]+p\left[y\right]\:$

$\mathrm{p3}≔\mathrm{v1}\left[x\right]+\mathrm{v2}\left[y\right]\:$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\,t\right]\,\mathrm{ranking}=\left[\left[\mathrm{v1}\,\mathrm{v2}\,p\right]\right]\right)$

$R≔\mathrm{PDE\_ring}$

$\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right]\,R\right)$

$\left[\mathrm{characterizable}\,\mathrm{characterizable}\,\mathrm{characterizable}\right]$

$S≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[t\,x\,y\right]\,\mathrm{ranking}=\left[\mathrm{lex}\left[p,\mathrm{v1},\mathrm{v2}\right]\right]\right)$

$S≔\mathrm{PDE\_ring}$

$J≔\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right]\,S\right)$

$J≔\left[\mathrm{characterizable}\right]$

$\mathrm{is\_orthonomic}\left(J\right)$

$\left[\mathrm{true}\right]$

$\mathrm{Rosenfeld\_Groebner}\left(\left[\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right]\,R\,J\right)$

$\left[\mathrm{characterizable}\right]$