notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of the first argument, p or L, is used.

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

Assume ideal is a regular differential chain and denote $R$ its embedding differential polynomial ring, and $\mathrm{I}$ the differential ideal that it defines. The function call NormalForm (p, ideal) returns a rational differential fraction, which is a normal form of p in $R$/$\mathrm{I}$. See the DifferentialAlgebra Glossary for further definitions.

The function call NormalForm (L, ideal) returns the list or the set of the normal forms of the elements of L with respect to ideal.

If ideal is a list of regular differential chains, then the function call NormalForm (p, ideal) returns the list of the normal forms of p with respect to all the chains, and, the function call NormalForm (L, ideal) returns the list of the normal forms of L with respect to all the chains.

If ideal is a differential polynomial ring, then the function call NormalForm (p, ideal) returns p, and, the function call NormalForm (L, ideal) returns L.

In the case of rational differential fractions, the computation may fail, making the function raise an error. These cases can completely be handled by using the casesplit=true option, documented below.

This command is part of the DifferentialAlgebra package. It can be called using the form NormalForm(...) after executing the command with(DifferentialAlgebra). It can also be directly called using the form DifferentialAlgebra[NormalForm](...).

The option casesplit = true permits to handle the cases which raise an error. There are, in this case, some restrictions on the function arguments:

The first argument, p, is restricted to a single rational differential fraction (lists and sets are not allowed).

The second argument, ideal is restricted to a single regular differential chain (lists of chains are not allowed).

The function call NormalForm (p, ideal, casesplit=true) returns a list [[[$F_1$, $C_1$], ..., [$F_n$, $C_n$]], [$C_{n+1}$, ..., $C_{n+p}$]]. The $C_i$ are regular differential chains. The intersection of the ideals that they define is equal to $\mathrm{I}$, in the differential case, and, more generally, whenever $\mathrm{I}$ is radical.

The denominator of p is regular modulo the ideals defined by the chains $C_1$, ..., $C_n$ and, for each $i$ in the range $\left[1\,n\right]$, the rational differential fraction $F_i$ is the normal form of p modulo $C_i$.

The denominator of p is zero modulo the ideals defined by the chains $C_{n+1}$, ..., $C_{n+p}$: these cases correspond to cases where the normal form does not exist.

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

$R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right]\,\mathrm{blocks}=\left[u\right]\right)$

$R≔\mathrm{differential\_ring}$

$\mathrm{NormalForm}\left(\left(u-1\right)\left(u+1\right)\,R\right)$

$u^2-1$

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[{u\left[t\right]}^2-4u\right]\,R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\,\mathrm{regular\_differential\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\right)$

$\left[\left[u_t^2-4u\right]\,\left[u\right]\right]$

$\mathrm{NormalForm}\left(u\left[t\right]\,\mathrm{ideal}\right)$

$\left[u_t\,0\right]$

$F≔\frac{u}{u\left[t\right]+u\left[t,t\right]}$

$F≔\frac{u}{u_t+u_{t,t}}$

$\mathrm{nf\_F}≔\mathrm{NormalForm}\left(F\,\mathrm{ideal}\left[1\right]\right)$

$\mathrm{nf\_F}≔\frac{u{u}_t-2u}{4u-4}$

$\mathrm{nf\_one\_over\_F}≔\mathrm{NormalForm}\left(\frac{1}{F}\,\mathrm{ideal}\left[1\right]\right)$

$\mathrm{nf\_one\_over\_F}≔\frac{u_t+2}{u}$

$\mathrm{one}≔\mathrm{normal}\left(\mathrm{nf\_F}\mathrm{nf\_one\_over\_F}\right)$

$\mathrm{one}≔\frac{\left(u_t-2\right)\left(u_t+2\right)}{4\left(u-1\right)}$

$\mathrm{NormalForm}\left(\mathrm{one}\,\mathrm{ideal}\left[1\right]\right)$

$1$

$\mathrm{ideal}≔\mathrm{Tools}:-\mathrm{PretendRegularDifferentialChain}\left(\left[{u\left[t\right]}^2-u^2\right]\,R\right)$

$\mathrm{ideal}≔\mathrm{regular\_differential\_chain}$

$\mathrm{res}≔\mathrm{NormalForm}\left(\frac{1}{{u\left[t\right]}^2-u^2}\,\mathrm{ideal}\,\mathrm{casesplit}=\mathrm{true}\right)$

$\mathrm{res}≔\left[\left[\right]\,\left[\mathrm{regular\_differential\_chain}\right]\right]$

$\mathrm{Equations}\left(\mathrm{res}\left[2,1\right]\right)$

$\left[-u^2+u_t^2\right]$

$\mathrm{res}≔\mathrm{NormalForm}\left(\frac{1}{u\left[t\right]-u}\,\mathrm{ideal}\,\mathrm{casesplit}=\mathrm{true}\right)$

$\mathrm{res}≔\left[\left[\left[-\frac{1}{2u}\,\mathrm{regular\_differential\_chain}\right]\right]\,\left[\mathrm{regular\_differential\_chain}\right]\right]$

$\mathrm{Equations}\left(\mathrm{res}\left[1,1,2\right]\right)$

$\left[u_t+u\right]$

$\mathrm{Equations}\left(\mathrm{res}\left[2,1\right]\right)$

$\left[u_t-u\right]$