diffalg(deprecated)/differential_sprem - Help

diffalg

 differential_sprem
 return sparse pseudo remainder of a differential polynomial

 Calling Sequence differential_sprem (q, L, R, 'h') differential_sprem (q, C, 'h')

Parameters

 q - differential polynomial in R L - list or a set of differential polynomials in R C - characterizable differential ideal R - differential polynomial ring h - (optional) name

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The function differential_sprem is an implementation of Ritt's reduction algorithm. It is an extension of the pseudo-remainder algorithm to differential polynomials.
 • L is assumed to form a differentially triangular set.
 • Let $A$ denote L or equations(C).
 • The function differential_sprem returns a differential polynomial r such that
 (a) $hq=\mathrm{mod}\left(r,\left[A\right]\right).$
 (b) No proper derivative of the leaders of the elements of $A$ appears in $r$.
 (c) The degree according to a leader of any element $a$ of $A$ is strictly less in $r$ than in $a$.
 (d) The differential polynomial h is a power product of factors of the  initials and the separants of the elements of A.
 • The differential_sprem(q, L, R, 'h') calling sequence returns an error message if $L$  contains 0. If $L$ contains a non zero element of the ground field of R, it returns zero.
 • The differential_sprem(q, C, 'h') calling sequence requires that q belong to the differential ring in which C is defined.
 The function rewrite_rules shows how the equations of C are interpreted by the pseudo-reduction algorithm.
 Then r is zero if and only if q belongs to C.
 • The command with(diffalg,differential_sprem) allows the use of the abbreviated form of this command.

Examples

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

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

Differential pseudo-division by a single differential polynomial:

 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x\right],\mathrm{ranking}=\left[u\right]\right):$
 > $p≔\left({u}_{[]}-1\right){u}_{x,x}+{u}_{x}$
 ${p}{≔}\left({{u}}_{{[}{]}}{-}{1}\right){}{{u}}_{{x}{,}{x}}{+}{{u}}_{{x}}$ (1)
 > $q≔\left({u}_{[]}^{2}-1\right){u}_{x}+1$
 ${q}{≔}\left({{u}}_{{[}{]}}^{{2}}{-}{1}\right){}{{u}}_{{x}}{+}{1}$ (2)
 > $r≔\mathrm{differential_sprem}\left(p,\left[q\right],R,'h'\right);$$h$
 ${r}{≔}{-}{{u}}_{{[}{]}}^{{3}}{-}{{u}}_{{[}{]}}^{{2}}{-}{{u}}_{{[}{]}}{+}{1}$
 $\left({{u}}_{{[}{]}}{+}{1}\right){}{\left({{u}}_{{[}{]}}^{{2}}{-}{1}\right)}^{{2}}$ (3)

Reduction according to a characterizable differential ideal:

 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[u\right]\right):$
 > $J≔\mathrm{Rosenfeld_Groebner}\left(\left[x{u}_{x,y}^{2}+y{u}_{y}+1\right],R\right)$
 ${J}{≔}\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (4)
 > $\mathrm{rewrite_rules}\left({J}_{1}\right)$
 $\left[{{u}}_{{x}{,}{y}}^{{2}}{=}{-}\frac{{y}{}{{u}}_{{y}}{+}{1}}{{x}}\right]$ (5)
 > $q≔{u}_{x,x,x,y,y}$
 ${q}{≔}{{u}}_{{x}{,}{x}{,}{x}{,}{y}{,}{y}}$ (6)
 > $r≔\mathrm{differential_sprem}\left(q,{J}_{1},'h'\right)$
 ${r}{≔}{-}{2}{}{x}{}{y}{}{{u}}_{{y}}{}{{u}}_{{x}{,}{y}}{+}{{y}}^{{2}}{}{{u}}_{{y}}{}{{u}}_{{y}{,}{y}}{+}{y}{}{{u}}_{{y}}^{{2}}{-}{2}{}{x}{}{{u}}_{{x}{,}{y}}{+}{y}{}{{u}}_{{y}{,}{y}}{+}{{u}}_{{y}}$ (7)
 > $h$
 $\frac{{8}}{{3}}{}{{x}}^{{4}}{}{{u}}_{{x}{,}{y}}^{{3}}$ (8)
 > $\mathrm{belongs_to}\left(hq-r,{J}_{1}\right)$
 ${\mathrm{true}}$ (9)