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DifferentialAlgebra[Tools]

 FieldElement
 decides membership in differential base fields

 Calling Sequence FieldElement (p, F, R, opts) FieldElement (L, F, R, opts)

Parameters

 p - a differential polynomial L - a list or a set of differential polynomials F - a field description R - a differential ring or ideal opts (optional) - a sequence of options

Options

 • 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).

Description

 • 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](...).

Examples

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

With no arguments, the field $k$ is the smallest field involving the rational numbers and the independent variables.

 > $\mathrm{FieldElement}\left(\left[1,t,u\right],\mathrm{field}\left(\right),R\right)$
 $\left[{\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{false}}\right]$ (2)

In this example, the field $k$ is the smallest differential field involving the rational numbers, the independent variables, and, the derivatives of $v$ and $w$.

 > $\mathrm{FieldElement}\left(\left[\frac{u}{{v}_{t}},{w}_{t,t},\frac{1}{v+w}\right],\mathrm{field}\left(\mathrm{generators}=\left[v,w\right]\right),R\right)$
 $\left[{\mathrm{false}}{,}{\mathrm{true}}{,}{\mathrm{true}}\right]$ (3)

In this example, the field $k$ is presented by generators and relations. The expression ${v}_{t,t}-2$ is $0$ in $k$.

 > $\mathrm{fieldrels}≔\mathrm{PretendRegularDifferentialChain}\left(\left[{v}_{t}^{2}-4v\right],R\right)$
 ${\mathrm{fieldrels}}{:=}{\mathrm{regular_differential_chain}}$ (4)
 > $\mathrm{NormalForm}\left({v}_{t,t}-2,\mathrm{fieldrels}\right)$
 ${0}$ (5)
 > $\mathrm{FieldElement}\left(\left({v}_{t,t}-2\right)u+{w}_{t}+1,\mathrm{field}\left(\mathrm{relations}=\mathrm{fieldrels},\mathrm{generators}=\left[v,w\right]\right),R\right)$
 ${\mathrm{true}}$ (6)