diffalg(deprecated)/reduced_form - Help

diffalg

 reduced_form
 compute a reduced form of a differential polynomial modulo a radical differential ideal

 Calling Sequence reduced_form (poly, R)

Parameters

 p - differential polynomial R - differential ring or radical differential ideal

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The function reduced_form returns a reduced form of p when R is a differential ring. When R is a radical differential ideal, it  returns a reduced form of p modulo R.
 • All the algorithms which return differential polynomials return them under reduced form, though  this is not a normal form of this polynomial.
 • The reduced form of p is a polynomial $r$ equivalent to p such that
 - All the monomials of r have non zero coefficients.
 - If the coefficients of p do not involve fractions, then the coefficients of r also do not.
 • Some extra simplifications may also happen, since the notation of the derivatives that appear in reduced forms is normalized.
 When using the jet notation, different names may represent the same derivative (for example, ${u}_{x,y}$ and ${u}_{y,x}$). The order of the derivations in the reduced form is given by the list of the derivations.
 When using the  diff notation, functions such as $u\left(x,y\right)$ and $u\left(y,x\right)$ are translated to the same differential indeterminate $u$. The order of the variables in the reduced form is given by the list of the derivations.
 • When  R is a differential ideal, a reduced form of p is a polynomial r equivalent to p modulo R. It satisfies $r=0$ if and only if p belongs to R.
 • The command with(diffalg,reduced_form) allows the use of the abbreviated form of this command.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

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

Illustration of reduced form with respect to a differential ring:

 > $F≔\mathrm{field_extension}\left(\mathrm{relations}=\left[{a}^{2}-b\right]\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{field_of_constants}=F,\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[u\right]\right):$
 > $p≔\left({a}^{2}-b\right){u}_{x}+{a}^{3}{u}_{x,y}-\frac{1{u}_{y,x}}{y}:$
 > $\mathrm{reduced_form}\left(p,R\right)$
 $\frac{{a}{}{b}{}{y}{}{{u}}_{{x}{,}{y}}{-}{{u}}_{{x}{,}{y}}}{{y}}$ (1)
 > $R≔\mathrm{differential_ring}\left(\mathrm{notation}=\mathrm{diff},\mathrm{derivations}=\left[x,y,z\right],\mathrm{ranking}=\left[u,v,w\right]\right):$
 > $p≔u\left(x,y,z\right)+u\left(y,x,z\right)+{\left(\frac{{\partial }^{2}}{\partial x\partial z}v\left(z,y,x\right)\right)}^{2}:$
 > $\mathrm{reduced_form}\left(p,R\right)$
 ${\left(\frac{{{\partial }}^{{2}}}{{\partial }{z}{}{\partial }{x}}{}{v}{}\left({x}{,}{y}{,}{z}\right)\right)}^{{2}}{+}{2}{}{u}{}\left({x}{,}{y}{,}{z}\right)$ (2)

Illustration of reduced form modulo a radical differential ideal:

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[\left[u,v\right]\right]\right):$
 > $\mathrm{p1}≔{v}_{[]}{u}_{x,x}-{u}_{x}:$
 > $\mathrm{p2}≔{u}_{x,y}:$
 > $\mathrm{p3}≔{u}_{y,y}^{2}-1:$
 > $P≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2},\mathrm{p3}\right],R\right)$
 ${P}{:=}\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (3)
 > $\mathrm{reduced_form}\left({u}_{x,x,x},P\right)$
 ${{u}}_{{x}{,}{x}{,}{x}}$ (4)
 > $\mathrm{reduced_form}\left({u}_{x,x,x}{v}_{y},P\right)$
 ${0}$ (5)
 > $\mathrm{reduced_form}\left({u}_{x,y}+{u}_{y,x},R\right)$
 ${2}{}{{u}}_{{x}{,}{y}}$ (6)