ReducedForm - Maple Help

DifferentialAlgebra

 ReducedForm
 computes reduced forms modulo regular differential chains

 Calling Sequence ReducedForm (p, ideal, opts) ReducedForm (L, ideal, opts)

Parameters

 p - a differential polynomial ideal - a differential polynomial ideal or ring L - a list or a set of differential polynomials opts (optional) - a sequence of options

Options

 • The opts arguments may contain one or more of the options below.
 • notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of the input first argument, p or L, is used.
 • memout = nonnegative. Specifies a memory limit, in MB, for the computation. Default is zero (no memory out).

Description

 • Assume ideal is a regular differential chain. Denote $R$ its embedding differential polynomial ring and $I$ the differential ideal that it defines.
 • The function call ReducedForm (p, ideal) returns a differential polynomial, which is a reduced form of p in $R$/$I$, as explained below.
 • The reduced form of a differential polynomial $p$, is a differential polynomial $q$, equivalent to $p$ modulo $I$. It is $0$ if $p$ belongs to $I$. If $p$ does not belong to $I$, its reduced form is obtained by applying some simplifications, using the regular differential chain, without introducing any rational fraction in the result.
 • The function call ReducedForm (L, ideal) returns the list or the set of the reduced forms of the elements of L with respect to ideal.
 • If ideal is a list of regular differential chains, then the function call ReducedForm (p, ideal) returns the list of the reduced forms of p with respect to all the chains, and, the function call ReducedForm (L, ideal) returns the list of the reduced forms of L with respect to all the chains.
 • If ideal is a differential polynomial ring, then the function call ReducedForm (p, ideal) returns p, and, the function call ReducedForm (L, ideal) returns L. In both cases, the differential polynomials are expanded.
 • This command is part of the DifferentialAlgebra package. It can be called using the form ReducedForm(...) after executing the command with(DifferentialAlgebra). It can also be directly called using the form DifferentialAlgebra[ReducedForm](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{blocks}=\left[w,\left[v,u\right],t\right]\right)$
 ${R}{≔}{\mathrm{differential_ring}}$ (1)
 > $\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[{u\left[x\right]}^{2}-4u,u\left[x,y\right]v\left[y\right]-u+1,v\left[x,x\right]-u\left[x\right]\right],R\right)$
 ${\mathrm{ideal}}{≔}\left[{\mathrm{regular_differential_chain}}\right]$ (2)
 > $\mathrm{Equations}\left(\mathrm{ideal},\mathrm{solved}\right)$
 $\left[\left[{{v}}_{{x}{,}{x}}{=}{{u}}_{{x}}{,}{{v}}_{{y}}{=}{-}\frac{{-}{u}{}{{u}}_{{x}}{}{{u}}_{{y}}{+}{{u}}_{{x}}{}{{u}}_{{y}}}{{4}{}{u}}{,}{{u}}_{{x}}^{{2}}{=}{4}{}{u}{,}{{u}}_{{y}}^{{2}}{=}{2}{}{u}\right]\right]$ (3)

The simplification by the above rewrite rules is not performed since it would give a rational differential fraction

 > $\mathrm{ReducedForm}\left(v\left[y\right],\mathrm{ideal}\right)$
 $\left[{{v}}_{{y}}\right]$ (4)

The simplification is performed, however, in this case

 > $\mathrm{ReducedForm}\left(uv\left[y\right],\mathrm{ideal}\right)$
 $\left[\frac{{1}}{{4}}{}{u}{}{{u}}_{{x}}{}{{u}}_{{y}}{-}\frac{{1}}{{4}}{}{{u}}_{{x}}{}{{u}}_{{y}}\right]$ (5)

In this case, the reduced form is obtained by simplifying partially the differential polynomial

 > $p≔\left(w+1\right)uv\left[y\right]+u\left[x,y\right]t$
 ${p}{≔}\left({w}{+}{1}\right){}{u}{}{{v}}_{{y}}{+}{{u}}_{{x}{,}{y}}{}{t}$ (6)
 > $\mathrm{rf}≔\mathrm{ReducedForm}\left(p,\mathrm{ideal}\left[1\right]\right)$
 ${\mathrm{rf}}{≔}\frac{{1}}{{4}}{}{w}{}{{u}}_{{x}}{}{{u}}_{{y}}{}{u}{-}\frac{{1}}{{4}}{}{w}{}{{u}}_{{x}}{}{{u}}_{{y}}{+}\frac{{1}}{{4}}{}{u}{}{{u}}_{{x}}{}{{u}}_{{y}}{+}{{u}}_{{x}{,}{y}}{}{t}{-}\frac{{1}}{{4}}{}{{u}}_{{x}}{}{{u}}_{{y}}$ (7)
 > $\mathrm{BelongsTo}\left(p-\mathrm{rf},\mathrm{ideal}\left[1\right]\right)$
 ${\mathrm{true}}$ (8)