InterReduce - Maple Help

Groebner

 InterReduce
 inter-reduce a list of polynomials

 Calling Sequence InterReduce(G, T, characteristic=p)

Parameters

 G - a list or set of polynomials T - a MonomialOrder or ShortMonomialOrder p - (optional) characteristic

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[{x}^{2}+xy-2,{x}^{2}-xy\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{x}{}{y}{-}{2}{,}{{x}}^{{2}}{-}{x}{}{y}\right]$ (1)
 > $\mathrm{LeadingMonomial}\left(F,\mathrm{tdeg}\left(x,y\right)\right)$
 $\left[{{x}}^{{2}}{,}{{x}}^{{2}}\right]$ (2)
 > $\mathrm{InterReduce}\left(F,\mathrm{tdeg}\left(x,y\right)\right)$
 $\left[{x}{}{y}{-}{1}{,}{{x}}^{{2}}{-}{1}\right]$ (3)
 > $r≔\mathrm{Reduce}\left(F\left[1\right],\left[F\left[2\right]\right],\mathrm{tdeg}\left(x,y\right)\right)$
 ${r}{≔}{x}{}{y}{-}{1}$ (4)
 > $\mathrm{Reduce}\left(F\left[2\right],\left[r\right],\mathrm{tdeg}\left(x,y\right)\right)$
 ${{x}}^{{2}}{-}{1}$ (5)

A set of inter-reduced (or autoreduced) polynomials is not a Groebner basis because syzygies are not considered.

 > $\mathrm{SPolynomial}\left(xy-1,{x}^{2}-1,\mathrm{tdeg}\left(x,y\right)\right)$
 ${-}{x}{+}{y}$ (6)
 > $\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(F,\mathrm{tdeg}\left(x,y\right)\right)$
 $\left[{x}{-}{y}{,}{{y}}^{{2}}{-}{1}\right]$ (7)

The next example is a non-commutative (Weyl) algebra where Dn*n = n*Dn + 1

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dn},n\right]\right)$
 ${A}{≔}{\mathrm{Ore_algebra}}$ (8)
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{Dn}\right)\right)$
 ${T}{≔}{\mathrm{monomial_order}}$ (9)
 > $\mathrm{w1}≔{n}^{2}{\mathrm{Dn}}^{2}+n$
 ${\mathrm{w1}}{≔}{{n}}^{{2}}{}{{\mathrm{Dn}}}^{{2}}{+}{n}$ (10)
 > $\mathrm{w2}≔{n}^{2}{\mathrm{Dn}}^{2}+\mathrm{Dn}$
 ${\mathrm{w2}}{≔}{{n}}^{{2}}{}{{\mathrm{Dn}}}^{{2}}{+}{\mathrm{Dn}}$ (11)
 > $\mathrm{InterReduce}\left(\left[\mathrm{w1},\mathrm{w2}\right],T\right)$
 $\left[{1}\right]$ (12)
 > $r≔\mathrm{Reduce}\left(\mathrm{w2},\left[\mathrm{w1}\right],T\right)$
 ${r}{≔}{\mathrm{Dn}}{-}{n}$ (13)
 > $\mathrm{Reduce}\left(\mathrm{w1},\left[r\right],T\right)$
 ${1}$ (14)