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

Characterizable and radical differential ideals are constructed by using the Rosenfeld_Groebner command. They are represented respectively by tables and list of tables.

A characterizable differential ideal is defined by a differential characteristic set.

The differential polynomials forming this characteristic set are accessed by equations. They are sorted by decreasing rank.

The inequations of a characterizable differential ideal consist of the factors of the initials and separants of the elements of its characteristic set.

If $C$ and $H$ are, respectively, the set of equations and inequations of the characterizable differential ideal J, then J is equal to the saturation differential ideal $\left[C\right]$:$H^{\mathrm{\infty }}$. It corresponds to the differential system $C=0,H\ne 0$.

A differential polynomial $p$  belongs to the characterizable differential ideal J if and only if $p$ is reduced to 0 by $C$ via differential_sprem.

The function rewrite_rules displays the equations of a characterizable differential ideal J as rewrite rules with the following the syntax:

$\mathrm{rank}\left(p\right)=-\frac{q}{\mathrm{initial}\left(p\right)}$, where, of course, $q=p-\mathrm{initial}\left(p\right)\mathrm{rank}\left(p\right)$.

(see rank, initial)

The list is sorted decreasingly.

If J is  a radical differential ideal given by a characteristic decomposition, that is, as a list of tables representing characterizable differential ideals, then the function is mapped on all its components.

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

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

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

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

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

$\mathrm{p1}≔v\left[\right]u\left[x,x\right]-u\left[x\right]\:$

$\mathrm{p2}≔u\left[x,y\right]\:$

$\mathrm{p3}≔{u\left[y,y\right]}^2-1\:$

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

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

$\mathrm{equations}\left(J\right)\;$$\mathrm{inequations}\left(J\right)$

$\left[\left[v\left[\right]u_{x,x}-u_x\,u_{x,y}\,u_{y,y}^2-1\,v_y\right]\,\left[u_x\,u_{y,y}^2-1\right]\right]$

$\left[\left[u_{y,y}\,v\left[\right]\right]\,\left[u_{y,y}\right]\right]$

$\mathrm{rewrite\_rules}\left(J\right)$

$\left[\left[u_{x,x}=\frac{u_x}{v\left[\right]}\,u_{x,y}=0\,u_{y,y}^2=1\,v_y=0\right]\,\left[u_x=0\,u_{y,y}^2=1\right]\right]$