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DifferentialAlgebra[Tools]

 DifferentialPrem
 the Ritt reduction algorithm

 Calling Sequence DifferentialPrem (p,regchain,opts) DifferentialPrem (p,redset,R,opts)

Parameters

 p - a differential polynomial regchain - a regular differential chain redset - a polynomial or 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.
 • reduction = full, partial or algebraic. Indicates the type of reduction to be performed. Depending on the type of reduction, the differential polynomial $r$ satisfies the following properties:
 – algebraic. For each leading rank ${v}^{d}$ of regchain, or, of redset, $r$ has degree less than $d$ in $v$.
 – partial. For each leading derivative $v$ of regchain, or, of redset, no proper derivative of $v$ occurs in $r$.
 – full. For each leading rank ${v}^{d}$ of regchain, or, of redset, no proper derivative of $v$ occurs in $r$, and, moreover, $r$ has degree less than $d$ in $v$.
 • notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of p is used.
 • memout = nonnegative. Specifies a memory limit, in MB, for the computation. Default is zero (no memory out).

Description

 • The function call DifferentialPrem (p,regchain) returns a sequence $h,r$ such that $h$ is a power product of initials and separants of regchain, $r$ is a differential polynomial fully reduced (see below) with respect to regchain, and, $hp=r$ modulo the differential ideal generated by the regular differential chain.
 • The function call DifferentialPrem (p,redset,R,opts) returns a sequence $h,r$ such that $h$ is a power product of initials and separants of redset, $r$ is a differential polynomial fully reduced with respect to each element of redset, and, $hp=r$ modulo the differential ideal generated by redset. The elements of redset must depend on at least, one derivative and have integer coefficients. All the differential polynomials are regarded as elements of R, or, of its embedding ring, if R is an ideal.
 • A differential polynomial p belongs to the differential ideal defined by regchain if and only if, the function call DifferentialPrem (p,regchain) returns a sequence $h,0$ whose second component is zero.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form DifferentialPrem(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][DifferentialPrem](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{blocks}=\left[\left[v,u\right]\right]\right)$
 ${R}{:=}{\mathrm{differential_ring}}$ (1)
 > $\mathrm{syst}≔\left[{u}_{x}^{2}-4u,{u}_{x,y}{v}_{y}-u+1,{v}_{x,x}-{u}_{x}\right]$
 ${\mathrm{syst}}{:=}\left[{{u}}_{{x}}^{{2}}{-}{4}{}{u}{,}{{u}}_{{x}{,}{y}}{}{{v}}_{{y}}{-}{u}{+}{1}{,}{{v}}_{{x}{,}{x}}{-}{{u}}_{{x}}\right]$ (2)
 > $\mathrm{ideal}≔{\mathrm{RosenfeldGroebner}\left(\mathrm{syst},R\right)}_{1}$
 ${\mathrm{ideal}}{:=}{\mathrm{regular_differential_chain}}$ (3)
 > $p≔{u}_{x,y}$
 ${p}{:=}{{u}}_{{x}{,}{y}}$ (4)
 > $h,r≔\mathrm{DifferentialPrem}\left(p,\mathrm{ideal}\right)$
 ${h}{,}{r}{:=}{2}{}{{u}}_{{y}}{,}{2}{}{{u}}_{{x}}$ (5)
 > $h,r≔\mathrm{DifferentialPrem}\left(p,\mathrm{syst},R\right)$
 ${h}{,}{r}{:=}{2}{}{{u}}_{{x}}{,}{4}{}{{u}}_{{y}}$ (6)

The ratio $r$/$h$ is equivalent to $p$ modulo the differential ideal defined by the regular differential chain, but, it is not the normal form of $p$

 > $\mathrm{NormalForm}\left(\frac{r}{h},\mathrm{ideal}\right),\mathrm{NormalForm}\left(p,\mathrm{ideal}\right)$
 $\frac{{1}}{{2}}{}\frac{{{u}}_{{x}}{}{{u}}_{{y}}}{{u}}{,}\frac{{1}}{{2}}{}\frac{{{u}}_{{x}}{}{{u}}_{{y}}}{{u}}$ (7)

Different modes of reduction are available

 > $h,r≔\mathrm{DifferentialPrem}\left({u}_{x}^{3},\mathrm{syst},R,\mathrm{reduction}=\mathrm{partial}\right)$
 ${h}{,}{r}{:=}{1}{,}{{u}}_{{x}}^{{3}}$ (8)
 > $h,r≔\mathrm{DifferentialPrem}\left({u}_{x,y}+{u}_{x}^{3},\mathrm{syst},\mathrm{reduction}=\mathrm{algebraic},R\right)$
 ${h}{,}{r}{:=}{{v}}_{{y}}{,}{4}{}{u}{}{{u}}_{{x}}{}{{v}}_{{y}}{+}{u}{-}{1}$ (9)