builds a regular differential chain - Maple Help

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DifferentialAlgebra[Tools][PretendRegularDifferentialChain] - builds a regular differential chain

 Calling Sequence PretendRegularDifferentialChain (eqns, R, opts)

Parameters

 eqns - a list or a set of differential rational fractions R - a differential polynomial ring or ideal opts (optional) - a sequence of options

Description

 • The function call PretendRegularDifferentialChain (eqns, R) builds a regular differential chain with the numerators of eqns, regarded as differential rational fractions of R, if R is a ring, or its embedding ring if R is an ideal. By default, the built regular differential chain is assumed to hold the attributes: differential, autoreduced, primitive, squarefree, normalized and coherent. For more details on attributes, see DifferentialAlgebra.
 • It is assumed that eqns already forms a regular differential chain with the above attributes. The list eqns does not need to be sorted.
 • In principle, the elements of eqns should be differential polynomials with integer numeric coefficients. However, rational differential fractions and expressions involving explicit relational operators, such as $p=q$ and $p\ne q$ are accepted. The rational differential fractions are replaced by their numerators. The expressions $p=q$ are converted into $p-q$. The expressions $p\ne q$ are ignored.
 • If eqns involves a parameter $p$, the equations stating that some derivatives of $p$ are zero, are automatically inserted in the regular differential chain, unless $p$, itself, is the leading derivative of some element of eqns.
 • If eqns is empty, the returned regular differential chain represents the zero ideal of R.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form PretendRegularDifferentialChain(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][PretendRegularDifferentialChain](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$
 > $R:=\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right],\mathrm{blocks}=\left[u,\left[s,c\right]\right]\right)$
 ${R}{:=}{\mathrm{differential_ring}}$ (1)

The function sorts the differential polynomials by increasing rank.

 > $\mathrm{ideal}:=\mathrm{PretendRegularDifferentialChain}\left(\left[{s}_{t}-c,{c}_{t}-s,{u}_{t}^{2}-su\right],R\right)$
 ${\mathrm{ideal}}{:=}{\mathrm{regular_differential_chain}}$ (2)
 > $\mathrm{Equations}\left(\mathrm{ideal}\right)$
 $\left[{-}{s}{}{u}{+}{{u}}_{{t}}^{{2}}{,}{{s}}_{{t}}{-}{c}{,}{{c}}_{{t}}{-}{s}\right]$ (3)
 > $\mathrm{Get}\left(\mathrm{attributes},\mathrm{ideal}\right)$
 $\left[{\mathrm{differential}}{,}{\mathrm{autoreduced}}{,}{\mathrm{primitive}}{,}{\mathrm{squarefree}}{,}{\mathrm{normalized}}\right]$ (4)

In the next example, the attribute normalized is omitted.

 > $\mathrm{ideal}:=\mathrm{PretendRegularDifferentialChain}\left(\left[c\left({s}_{t}-c\right),{\left({c}_{t}-s\right)}^{2},{u}_{t}^{2}-{c}_{t}u\right],\mathrm{attributes}=\left[\mathrm{differential},\mathrm{autoreduced},\mathrm{primitive}\right],\mathrm{pretend}=\mathrm{false},R\right)$
 ${\mathrm{ideal}}{:=}{\mathrm{regular_differential_chain}}$ (5)
 > $\mathrm{Equations}\left(\mathrm{ideal}\right)$
 $\left[{-}{s}{}{u}{+}{{u}}_{{t}}^{{2}}{,}{{s}}_{{t}}{-}{c}{,}{{c}}_{{t}}{-}{s}\right]$ (6)
 > $\mathrm{Get}\left(\mathrm{attributes},\mathrm{ideal}\right)$
 $\left[{\mathrm{differential}}{,}{\mathrm{autoreduced}}{,}{\mathrm{primitive}}{,}{\mathrm{squarefree}}\right]$ (7)