diffalg(deprecated)/reduced - Maple Help

diffalg

 reduced
 test if a differential polynomial is reduced with respect to a set of differential polynomials

 Calling Sequence reduced (p, F, R, code) reduced (p, P, code)

Parameters

 p - differential polynomial in R F - differential polynomial or list/set of differential polynomials in R R - differential polynomial ring code - (optional) name; 'fully' or 'partially' P - characterizable differential ideal

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The function reduced returns true if p is reduced with respect to F  or with respect to the equations of P.  It returns false otherwise.
 • A differential polynomial p is said to be partially reduced with respect to a polynomial q if no proper derivative of the leader of q appears in p.
 A differential polynomial p is said to be fully reduced with respect to a polynomial q if it is partially reduced with respect to q and if its degree in the leader of q is less than the degree of q in this leader.
 • A differential polynomial p is said to be reduced with respect to a set of differential polynomials F if it is reduced with respect to each element of F.
 • If code is omitted, it is assumed to be 'fully'.
 • If the second form of the function is used and P is a radical differential ideal defined  by a list of characterizable differential ideals then the function is mapped over all the components of the ideal.
 • The command with(diffalg,reduced) allows the use of the abbreviated form of this command.

Examples

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

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[u\right]\right)$
 ${R}{≔}{\mathrm{PDE_ring}}$ (1)
 > $q≔{u\left[x\right]}^{2}-4u\left[\right]$
 ${q}{≔}{{u}}_{{x}}^{{2}}{-}{4}{}{u}\left[\right]$ (2)
 > $\mathrm{reduced}\left({u\left[x\right]}^{3},q,R\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{reduced}\left({u\left[x\right]}^{3},q,R,'\mathrm{partially}'\right)$
 ${\mathrm{true}}$ (4)
 > $P≔\mathrm{Rosenfeld_Groebner}\left(\left[q\right],R\right)$
 ${P}{≔}\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (5)
 > $\mathrm{equations}\left(P\right)$
 $\left[\left[{{u}}_{{x}}^{{2}}{-}{4}{}{u}\left[\right]\right]{,}\left[{u}\left[\right]\right]\right]$ (6)
 > $\mathrm{reduced}\left(u\left[x\right],P\right)$
 $\left[{\mathrm{true}}{,}{\mathrm{false}}\right]$ (7)