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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

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

withDifferentialAlgebra:withTools:

R:=DifferentialRingderivations=t,blocks=u,v,w

R:=differential_ring

(1)

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

FieldElement1,t,u,field,R

true,true,false

(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.

FieldElementuvt,wt,t,1v+w,fieldgenerators=v,w,R

false,true,true

(3)

In this example, the field k is presented by generators and relations. The expression vt,t2 is 0 in k.

fieldrels:=PretendRegularDifferentialChainvt24v,R

fieldrels:=regular_differential_chain

(4)

NormalFormvt,t2,fieldrels

0

(5)

FieldElementvt,t2u+wt+1,fieldrelations=fieldrels,generators=v,w,R

true

(6)

See Also

DifferentialAlgebra, Tools, RosenfeldGroebner


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