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

The function belongs_to returns true if q belongs to J. Otherwise, false is returned.

Mathematically, q belongs to J if and only if q vanishes on all the zeros of  J.

The differential polynomial q belongs to J if and only if it belongs to all the components of the characteristic decomposition.

q belongs to a characterizable component $J_j$ of J if and only if the differential remainder of q by the differential characteristic set defining  $J_j$ is zero.

Characteristic decomposition of radical differential ideal are computed by Rosenfeld_Groebner.

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

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

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

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

$\mathrm{p2}≔u\left[x,y\right]\:$

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

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

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

$\mathrm{belongs\_to}\left(v\left[y\right]\,J\left[1\right]\right),\mathrm{belongs\_to}\left(v\left[y\right]\,J\left[2\right]\right),\mathrm{belongs\_to}\left(v\left[y\right]\,J\right)$

$\mathrm{true},\mathrm{false},\mathrm{false}$

$\mathrm{belongs\_to}\left(u\left[x\right]\,J\left[1\right]\right),\mathrm{belongs\_to}\left(u\left[x\right]\,J\left[2\right]\right),\mathrm{belongs\_to}\left(u\left[x\right]\,J\right)$

$\mathrm{false},\mathrm{true},\mathrm{false}$

$\mathrm{belongs\_to}\left(u\left[x\right]v\left[y\right]\,J\right)$

$\mathrm{true}$