The InterReduce command inter-reduces a list or set of polynomials G with respect to a monomial order T. The result is a list of polynomials defining the same ideal as G, but where no term of a polynomial is reducible by the leading term of another polynomial.  See also the help page for Groebner[Reduce]. The resulting list is sorted in ascending order of leading monomial.

A typical use of this command is to construct a reduced Groebner basis from a Groebner basis computed outside of Maple. See the Monomial Orders help page for more information about the monomial orders that are available in Maple.

If T is a ShortMonomialOrder then the elements of G must be polynomials in the ring implied by T.  If T is a MonomialOrder created with the Groebner[MonomialOrder] command, then the elements of G must be members of the algebra used to define T.  

The optional argument characteristic=p can be used to specify the ring characteristic when T is a ShortMonomialOrder. The default value is zero.

Note that the inter_reduce command is deprecated.  It may not be supported in a future Maple release.

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

$F≔\left[x^2+xy-2\,x^2-xy\right]$

$F≔\left[x^2+xy-2\,x^2-xy\right]$

$\mathrm{LeadingMonomial}\left(F\,\mathrm{tdeg}\left(x\,y\right)\right)$

$\left[x^2\,x^2\right]$

$\mathrm{InterReduce}\left(F\,\mathrm{tdeg}\left(x\,y\right)\right)$

$\left[xy-1\,x^2-1\right]$

$r≔\mathrm{Reduce}\left(F\left[1\right]\,\left[F\left[2\right]\right]\,\mathrm{tdeg}\left(x\,y\right)\right)$

$r≔xy-1$

$\mathrm{Reduce}\left(F\left[2\right]\,\left[r\right]\,\mathrm{tdeg}\left(x\,y\right)\right)$

$x^2-1$

$\mathrm{SPolynomial}\left(xy-1\,x^2-1\,\mathrm{tdeg}\left(x\,y\right)\right)$

$-x+y$

$\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(F\,\mathrm{tdeg}\left(x\,y\right)\right)$

$\left[x-y\,y^2-1\right]$

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\:$

$A≔\mathrm{diff\_algebra}\left(\left[\mathrm{Dn}\,n\right]\right)$

$A≔\mathrm{Ore\_algebra}$

$T≔\mathrm{MonomialOrder}\left(A\,\mathrm{tdeg}\left(\mathrm{Dn}\right)\right)$

$T≔\mathrm{monomial\_order}$

$\mathrm{w1}≔n^2{\mathrm{Dn}}^2+n$

$\mathrm{w1}≔n^2{\mathrm{Dn}}^2+n$

$\mathrm{w2}≔n^2{\mathrm{Dn}}^2+\mathrm{Dn}$

$\mathrm{w2}≔n^2{\mathrm{Dn}}^2+\mathrm{Dn}$

$\mathrm{InterReduce}\left(\left[\mathrm{w1}\,\mathrm{w2}\right]\,T\right)$

$\left[1\right]$

$r≔\mathrm{Reduce}\left(\mathrm{w2}\,\left[\mathrm{w1}\right]\,T\right)$

$r≔\mathrm{Dn}-n$

$\mathrm{Reduce}\left(\mathrm{w1}\,\left[r\right]\,T\right)$

$1$