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diffalg

 belongs_to
 test if a differential polynomial belongs to a radical differential ideal

 Calling Sequence belongs_to (q, J)

Parameters

 q - differential polynomial J -

Description

 • 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.

Examples

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

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[\left[u,v\right]\right]\right):$
 > $\mathrm{p1}≔{v}_{[]}{u}_{x,x}-{u}_{x}:$
 > $\mathrm{p2}≔{u}_{x,y}:$
 > $\mathrm{p3}≔{u}_{y,y}^{2}-1:$
 > $J≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2},\mathrm{p3}\right],R\right)$
 ${J}{≔}\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (1)
 > $\mathrm{belongs_to}\left({v}_{y},{J}_{1}\right),\mathrm{belongs_to}\left({v}_{y},{J}_{2}\right),\mathrm{belongs_to}\left({v}_{y},J\right)$
 ${\mathrm{true}}{,}{\mathrm{false}}{,}{\mathrm{false}}$ (2)
 > $\mathrm{belongs_to}\left({u}_{x},{J}_{1}\right),\mathrm{belongs_to}\left({u}_{x},{J}_{2}\right),\mathrm{belongs_to}\left({u}_{x},J\right)$
 ${\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{false}}$ (3)
 > $\mathrm{belongs_to}\left({u}_{x}{v}_{y},J\right)$
 ${\mathrm{true}}$ (4)