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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

 • 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).

Description

 • 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 $I$ 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 $I$. 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](...).

Examples

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

Basic illustration

 • The following examples illustrate the function, syntactically.
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right],\mathrm{blocks}=\left[u,\left[s,c\right]\right]\right)$
 ${R}{≔}{\mathrm{differential_ring}}$ (1)
 > $\mathrm{ideal}≔\mathrm{PretendRegularDifferentialChain}\left(\left[{c}^{2}+{s}^{2}-1,c\left[t\right]+s,u\right],R,\mathrm{pretend}=\mathrm{false}\right)$
 ${\mathrm{ideal}}{≔}{\mathrm{regular_differential_chain}}$ (2)
 > $\mathrm{Equations}\left(\mathrm{ideal}\right)$
 $\left[{u}{,}{{c}}_{{t}}{+}{s}{,}{{c}}^{{2}}{+}{{s}}^{{2}}{-}{1}\right]$ (3)
 > $f≔\left(c\left[t\right]+s\right)u\left[t\right]+{u\left[t\right]}^{2}$
 ${f}{≔}\left({{c}}_{{t}}{+}{s}\right){}{{u}}_{{t}}{+}{{u}}_{{t}}^{{2}}$ (4)
 > $\mathrm{prepeq}≔\mathrm{PreparationEquation}\left(f,\mathrm{ideal}\right)$
 ${\mathrm{prepeq}}{≔}{{u}}_{{t}}{}{s}{+}{{u}}_{{t}}{}{{c}}_{{t}}{+}{{u}}_{{t}}^{{2}}{=}{\mathrm{z2}}{}{{\mathrm{z3}}}_{{t}}{+}{{\mathrm{z3}}}_{{t}}^{{2}}$ (5)
 • If one substitutes the ${A}_{i}$ to the ${z}_{i}$, the equation becomes an equality.
 > $\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{prepeq},\left[\mathrm{z3}=u,\mathrm{z2}=c\left[t\right]+s\right]\right)\right)$
 ${{u}}_{{t}}{}{s}{+}{{u}}_{{t}}{}{{c}}_{{t}}{+}{{u}}_{{t}}^{{2}}{=}{{u}}_{{t}}{}{s}{+}{{u}}_{{t}}{}{{c}}_{{t}}{+}{{u}}_{{t}}^{{2}}$ (6)
 • Changing the identifier for the ${z}_{i}$.
 > $\mathrm{PreparationEquation}\left(f,\mathrm{ideal},\mathrm{zstring}="A\left(%d\right)"\right)$
 ${{u}}_{{t}}{}{s}{+}{{u}}_{{t}}{}{{c}}_{{t}}{+}{{u}}_{{t}}^{{2}}{=}{{A}{}\left({3}\right)}_{{t}}^{{2}}{+}{A}{}\left({2}\right){}{{A}{}\left({3}\right)}_{{t}}$ (7)
 • Since all monomials ${t}_{i}$ have degree $q=2$, the preparation congruence is equal to the preparation equation.
 > $\mathrm{PreparationEquation}\left(f,\mathrm{ideal},\mathrm{congruence}=\mathrm{true}\right)$
 ${{u}}_{{t}}{}{s}{+}{{u}}_{{t}}{}{{c}}_{{t}}{+}{{u}}_{{t}}^{{2}}{=}{\mathrm{z2}}{}{{\mathrm{z3}}}_{{t}}{+}{{\mathrm{z3}}}_{{t}}^{{2}}$ (8)
 • However, if the two first elements of the regular differential chain are considered as base field defining equations, then, only one monomial ${t}_{i}$ is left in the congruence.
 > $\mathrm{PreparationEquation}\left(f,\mathrm{ideal},\mathrm{congruence}=\mathrm{true},n=2\right)$
 ${{u}}_{{t}}{}{s}{+}{{u}}_{{t}}{}{{c}}_{{t}}{+}{{u}}_{{t}}^{{2}}{=}{{\mathrm{z3}}}_{{t}}^{{2}}$ (9)

The Low Power Theorem

 • The next example illustrates the Low Power Theorem. See [R50, chapter III] and [K73, chapter IV, section 15].
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right],\mathrm{blocks}=\left[y\right]\right)$
 ${R}{≔}{\mathrm{differential_ring}}$ (10)
 > $f≔{y\left[t\right]}^{3}-4y\left[t\right]yt+8{y}^{2}$
 ${f}{≔}{-}{4}{}{{y}}_{{t}}{}{y}{}{t}{+}{{y}}_{{t}}^{{3}}{+}{8}{}{{y}}^{{2}}$ (11)
 • First compute a representation of the radical of the differential ideal generated by $f$, by means of RosenfeldGroebner.
 > $\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[f\right],R\right)$
 ${\mathrm{ideal}}{≔}\left[{\mathrm{regular_differential_chain}}{,}{\mathrm{regular_differential_chain}}{,}{\mathrm{regular_differential_chain}}\right]$ (12)
 > $\mathrm{Equations}\left(\mathrm{ideal}\right)$
 $\left[\left[{-}{4}{}{{y}}_{{t}}{}{y}{}{t}{+}{{y}}_{{t}}^{{3}}{+}{8}{}{{y}}^{{2}}\right]{,}\left[{-}{4}{}{{t}}^{{3}}{+}{27}{}{y}\right]{,}\left[{y}\right]\right]$ (13)
 • 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 $f$, with respect to each of the two singular components, i.e., the two last ones.
 • In the first case, there is only one monomial ${t}_{1}$, of the form ${z}_{1}^{q}$. Thus this regular differential chain must be kept in the decomposition.
 > $\mathrm{rhs}\left(\mathrm{PreparationEquation}\left(f,\mathrm{ideal}\left[2\right],\mathrm{congruence}=\mathrm{true}\right)\right)$
 ${167365651248}{}{{t}}^{{3}}{}{\mathrm{z1}}$ (14)
 • In the second case, the right hand-side of the preparation congruence involves two monomials. Thus this regular differential chain is redundant.
 > $\mathrm{rhs}\left(\mathrm{PreparationEquation}\left(f,\mathrm{ideal}\left[3\right],\mathrm{congruence}=\mathrm{true}\right)\right)$
 ${-}{4}{}{t}{}{\mathrm{z1}}{}{{\mathrm{z1}}}_{{t}}{+}{8}{}{{\mathrm{z1}}}^{{2}}$ (15)
 • Indeed, RosenfeldGroebner with the option singsol = essential removes the second singular component from the decomposition.
 > $\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(f,R,\mathrm{singsol}=\mathrm{essential}\right)$
 ${\mathrm{ideal}}{≔}\left[{\mathrm{regular_differential_chain}}{,}{\mathrm{regular_differential_chain}}\right]$ (16)
 > $\mathrm{Equations}\left(\mathrm{ideal}\right)$
 $\left[\left[{-}{4}{}{{y}}_{{t}}{}{y}{}{t}{+}{{y}}_{{t}}^{{3}}{+}{8}{}{{y}}^{{2}}\right]{,}\left[{-}{4}{}{{t}}^{{3}}{+}{27}{}{y}\right]\right]$ (17)