The opts arguments may contain one or more of the options below.

notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of the first argument is used.

memout = nonnegative. Specifies a memory limit, in MB, for the computation. Default is zero (no memory out).

The function call FieldElement (p,F,R) returns true if the differential polynomial p belongs to the differential field $k$ defined F and R, else it returns false. The differential polynomial p is regarded as a differential polynomial of R, if R is a differential ring, or, of the embedding ring of R, if R is an ideal.

The argument F has the form field (generators = G, relations = regchain). It defines a differential field $k$ presented by the list of derivatives G and the regular differential chain regchain. The field $k$ contains the rational numbers. Its set of generators is made of the independent variables, plus all the dependent variables occurring in G or in the differential polynomials of regchain. Every polynomial expression belonging to the differential ideal defined by regchain, is zero in $k$. Every other polynomial expression between the generators of $k$, is invertible in $k$. Notes:

Any of the arguments generators = G, and, relations = regchain can be omitted.

It is required that the generators of $k$ appear at the bottom of the ranking of R, and, that any block which involves a generator of $k$, purely consists of generators of $k$.

The function call FieldElement (L,F,R) returns a list or a set of boolean.

This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form FieldElement(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][FieldElement](...).

$\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right)\:$$\mathrm{with}\left(\mathrm{Tools}\right)\:$

$R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right]\,\mathrm{blocks}=\left[u\,v\,w\right]\right)$

$R≔\mathrm{differential\_ring}$

$\mathrm{FieldElement}\left(\left[1\,t\,u\right]\,\mathrm{field}\left(\right)\,R\right)$

$\left[\mathrm{true}\,\mathrm{true}\,\mathrm{false}\right]$

$\mathrm{FieldElement}\left(\left[\frac{u}{v\left[t\right]}\,w\left[t,t\right]\,\frac{1}{v+w}\right]\,\mathrm{field}\left(\mathrm{generators}=\left[v\,w\right]\right)\,R\right)$

$\left[\mathrm{false}\,\mathrm{true}\,\mathrm{true}\right]$

$\mathrm{fieldrels}≔\mathrm{PretendRegularDifferentialChain}\left(\left[{v\left[t\right]}^2-4v\right]\,R\right)$

$\mathrm{fieldrels}≔\mathrm{regular\_differential\_chain}$

$\mathrm{NormalForm}\left(v\left[t,t\right]-2\,\mathrm{fieldrels}\right)$

$0$

$\mathrm{FieldElement}\left(\left(v\left[t,t\right]-2\right)u+w\left[t\right]+1\,\mathrm{field}\left(\mathrm{relations}=\mathrm{fieldrels}\,\mathrm{generators}=\left[v\,w\right]\right)\,R\right)$

$\mathrm{true}$