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.

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 .

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.

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

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](...).

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

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.

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.

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".

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.

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