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

The function differential_sprem is an implementation of Ritt's reduction algorithm. It is an extension of the pseudo-remainder algorithm to differential polynomials.

L is assumed to form a differentially triangular set.

Let $A$ denote L or equations(C).

The function differential_sprem returns a differential polynomial r such that

(a) $hq\=\mathrm{mod}\left(r\,\left[A\right]\right)\.$ 

(b) No proper derivative of the leaders of the elements of $A$ appears in $r$.

(c) The degree according to a leader of any element $a$ of $A$ is strictly less in $r$ than in $a$.

(d) The differential polynomial h is a power product of factors of the  initials and the separants of the elements of A.

The differential_sprem(q, L, R, 'h') calling sequence returns an error message if $L$  contains 0. If $L$ contains a non zero element of the ground field of R, it returns zero.

The differential_sprem(q, C, 'h') calling sequence requires that q belong to the differential ring in which C is defined.

The function rewrite_rules shows how the equations of C are interpreted by the pseudo-reduction algorithm.

Then r is zero if and only if q belongs to C.

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

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

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

$p≔\left(u\left[\right]-1\right)u\left[x,x\right]+u\left[x\right]$

$p≔\left(u\left[\right]-1\right)u_{x,x}+u_x$

$q≔\left({u\left[\right]}^2-1\right)u\left[x\right]+1$

$q≔\left({u\left[\right]}^2-1\right)u_x+1$

$r≔\mathrm{differential\_sprem}\left(p\,\left[q\right]\,R\,'h'\right)\;$$h$

$r≔-{u\left[\right]}^3-{u\left[\right]}^2-u\left[\right]+1$

$\left(u\left[\right]+1\right){\left({u\left[\right]}^2-1\right)}^2$

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

$J≔\mathrm{Rosenfeld\_Groebner}\left(\left[x{u\left[x,y\right]}^2+yu\left[y\right]+1\right]\,R\right)$

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

$\mathrm{rewrite\_rules}\left(J\left[1\right]\right)$

$\left[u_{x,y}^2=-\frac{y{u}_y+1}{x}\right]$

$q≔u\left[x,x,x,y,y\right]$

$q≔u_{x,x,x,y,y}$

$r≔\mathrm{differential\_sprem}\left(q\,J\left[1\right]\,'h'\right)$

$r≔-2xy{u}_y{u}_{x,y}+y^2{u}_y{u}_{y,y}+y{u}_y^2-2x{u}_{x,y}+y{u}_{y,y}+u_y$

$h$

$\frac{8x^4{u}_{x,y}^3}{3}$

$\mathrm{belongs\_to}\left(hq-r\,J\left[1\right]\right)$

$\mathrm{true}$