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.

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

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[u\right]\right)$

$R≔\mathrm{PDE\_ring}$

$q≔{u\left[x\right]}^2-4u\left[\right]$

$q≔u_x^2-4u\left[\right]$

$\mathrm{reduced}\left({u\left[x\right]}^3\,q\,R\right)$

$\mathrm{false}$

$\mathrm{reduced}\left({u\left[x\right]}^3\,q\,R\,'\mathrm{partially}'\right)$

$\mathrm{true}$

$P≔\mathrm{Rosenfeld\_Groebner}\left(\left[q\right]\,R\right)$

$P≔\left[\mathrm{characterizable}\,\mathrm{characterizable}\right]$

$\mathrm{equations}\left(P\right)$

$\left[\left[u_x^2-4u\left[\right]\right]\,\left[u\left[\right]\right]\right]$

$\mathrm{reduced}\left(u\left[x\right]\,P\right)$

$\left[\mathrm{true}\,\mathrm{false}\right]$