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DifferentialAlgebra

 BelongsTo
 decides membership in differential ideals

 Calling Sequence BelongsTo(p, ideal, opts) BelongsTo(L, ideal, opts)

Parameters

 p - a differential polynomial L - a list or a set of differential polynomials ideal - a differential ideal opts (optional) - a sequence of options

Description

 • The function call BelongsTo(p,ideal) returns true if the differential polynomial p belongs to the differential ideal represented by ideal, else it returns false.
 • If ideal is a list of regular differential chains, the function returns true if and only if p belongs to all the elements of the list. If the first argument, L, is a list or a set off differential polynomials, the call BelongsTo(L, ideal) returns a list or a set of true / false.
 • This command is part of the DifferentialAlgebra package. It can be called using the form BelongsTo(...) after executing the command with(DifferentialAlgebra). It can also be directly called using the form DifferentialAlgebra[BelongsTo](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right],\mathrm{blocks}=\left[u\right]\right)$
 ${R}{:=}{\mathrm{differential_ring}}$ (1)

Every differential polynomial belongs to the unit differential ideal

 > $\mathrm{BelongsTo}\left({u}_{t},\left[\right]\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[{u}_{t}^{2}-4u\right],R\right)$
 ${\mathrm{ideal}}{:=}\left[{\mathrm{regular_differential_chain}}{,}{\mathrm{regular_differential_chain}}\right]$ (3)

The two first differential polynomials do not belong to ideal but their product does.

 > $\mathrm{BelongsTo}\left(\left[{u}_{t},{u}_{t,t}-2,{u}_{t}\left({u}_{t,t}-2\right)\right],\mathrm{ideal}\right)$
 $\left[{\mathrm{false}}{,}{\mathrm{false}}{,}{\mathrm{true}}\right]$ (4)