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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

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

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

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

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

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

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