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Calling Sequence
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PreparationEquation (f, regchain, opts)
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Parameters
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f
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a differential polynomial
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regchain
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a regular differential chain
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opts (optional)
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a sequence of options
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Options
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The opts arguments may contain one or more of the options below.
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congruence = true. In the right hand-side of the returned preparation equation, denote the minimum total degree of the monomials . All the terms such that the total degree of is greater than , are removed from the right hand-side of the preparation equation. This stripped preparation equation is called a preparation congruence of f.
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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).
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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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Let denote the differential ideal defined by regchain and denote , ..., the differential polynomials which constitute the chain. Introduce new dependent variables . Each variable represents the differential polynomial .
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The returned preparation equation is an expression having the form = + ... + . The differential polynomial is a power product of initials and separants of the . The coefficients are reduced and regular with respect to . The monomials are power products of the variables and their derivatives. They satisfy some further properties, described in [K73, chapter IV, section 13]. If each is replaced by the corresponding , in all terms , then the preparation equation becomes a true equality.
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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](...).
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Examples
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Basic illustration
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The following examples illustrate the function, syntactically.
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If one substitutes the to the , the equation becomes an equality.
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Changing the identifier for the .
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Since all monomials have degree , the preparation congruence is equal to the preparation equation.
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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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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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First compute a representation of the radical of the differential ideal generated by , by means of RosenfeldGroebner.
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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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In the second case, the right hand-side of the preparation congruence involves two monomials. Thus this regular differential chain is redundant.
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Indeed, RosenfeldGroebner with the option singsol = essential removes the second singular component from the decomposition.
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