DifferentialAlgebra[Tools][PreparationEquation]  returns the preparation equation of a differential polynomial

Calling Sequence


PreparationEquation (f, regchain, opts)


Parameters


f



a differential polynomial

regchain



a regular differential chain

opts (optional)



a sequence of options





Options


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

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n = nonnegative (default value is ). This option is useful in conjunction with the option congruence = true. The n first differential polynomials , ..., of regchain are considered as equations defining the base field of f, and, of the differential polynomials , ..., . Reductions by the base field equations are not taken into account for computing the preparation congruence of f: the terms involving derivatives of , ..., are not considered for determining , and, do not appear in the preparation congruence.

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zstring = string. This option permits to customize the identifier used for the new variables . It must be a valid MAPLE identifier (possibly an indexed) involving the substring "%d".

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

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



Description


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The function call PreparationEquation (f, regchain) returns a preparation equation [K73, chapter IV, section 13] for f with respect to regchain. The argument f is regarded as a differential polynomial of the embedding ring of regchain.

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Preparation equations are an important tool in the context of the Low Power Theorem. See RosenfeldGroebner with the option singsol = essential.

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



Examples


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Basic illustration


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The following examples illustrate the function, syntactically.

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 (1) 
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 (2) 
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 (3) 
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 (5) 
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If one substitutes the to the , the equation becomes an equality.

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 (6) 
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Changing the identifier for the .

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 (7) 
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Since all monomials have degree , the preparation congruence is equal to the preparation equation.

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 (8) 
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However, if the two first elements of the regular differential chain are considered as base field defining equations, then, only one monomial is left in the congruence.

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 (9) 


The Low Power Theorem


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The next example illustrates the Low Power Theorem. See [R50, chapter III] and [K73, chapter IV, section 15].

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 (10) 
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 (11) 
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First compute a representation of the radical of the differential ideal generated by , by means of RosenfeldGroebner.

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 (12) 
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 (13) 
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Second, remove any regular differential chain which involve two or more differential polynomials, by application of the Component Theorem [K73, chapter IV, section 14]. In our case, no regular differential chain is removed by this process. Third, compute a preparation congruence for , with respect to each of the two singular components, i.e., the two last ones.

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In the first case, there is only one monomial , of the form . Thus this regular differential chain must be kept in the decomposition.

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 (14) 
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In the second case, the right handside of the preparation congruence involves two monomials. Thus this regular differential chain is redundant.

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 (15) 
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Indeed, RosenfeldGroebner with the option singsol = essential removes the second singular component from the decomposition.

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 (16) 
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 (17) 


