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 $q$ the minimum total degree of the monomials $t_i$. All the terms $c_i{t}_i$ such that the total degree of $t_i$ is greater than $q$, 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 $0$). This option is useful in conjunction with the option congruence = true. The n first differential polynomials $A_1$, ..., $A_n$ of regchain are considered as equations defining the base field of f, and, of the differential polynomials $A_{n+1}$, ..., $A_r$. Reductions by the base field equations are not taken into account for computing the preparation congruence of f: the terms $t_i$ involving derivatives of $z_1$, ..., $z_n$ are not considered for determining $q$, and, do not appear in the preparation congruence.

zstring = string. This option permits to customize the identifier used for the new variables $z_k$. 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).

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 $\mathrm{I}$ denote the differential ideal defined by regchain and denote $A_1$, ..., $A_r$ the differential polynomials which constitute the chain. Introduce $r$ new dependent variables $z_i$. Each variable $z_i$ represents the differential polynomial $A_i$.

The returned preparation equation is an expression having the form $hf$ = $c_1{t}_1$ + ... + $c_n{t}_n$. The differential polynomial $h$ is a power product of initials and separants of the $A_k$. The coefficients $c_i$ are reduced and regular with respect to $\mathrm{I}$. The monomials $t_i$ are power products of the $z_k$ variables and their derivatives. They satisfy some further properties, described in [K73, chapter IV, section 13]. If each $z_k$ is replaced by the corresponding $A_k$, in all terms $t_i$, 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](...).

$\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right)\:$$\mathrm{with}\left(\mathrm{Tools}\right)\:$