Groebner
InterReduce
inter-reduce a list of polynomials
Calling Sequence
Parameters
Description
Examples
InterReduce(G, T, characteristic=p)
G
-
a list or set of polynomials
T
a MonomialOrder or ShortMonomialOrder
p
(optional) characteristic
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.
withGroebner:
F≔x2+xy−2,x2−xy
LeadingMonomialF,tdegx,y
x2,x2
InterReduceF,tdegx,y
xy−1,x2−1
r≔ReduceF1,F2,tdegx,y
r≔xy−1
ReduceF2,r,tdegx,y
x2−1
A set of inter-reduced (or autoreduced) polynomials is not a Groebner basis because syzygies are not considered.
SPolynomialxy−1,x2−1,tdegx,y
−x+y
GroebnerBasisF,tdegx,y
x−y,y2−1
The next example is a non-commutative (Weyl) algebra where Dn*n = n*Dn + 1
withOre_algebra:
A≔diff_algebraDn,n
A≔Ore_algebra
T≔MonomialOrderA,tdegDn
T≔monomial_order
w1≔n2Dn2+n
w2≔n2Dn2+Dn
InterReducew1,w2,T
1
r≔Reducew2,w1,T
r≔Dn−n
Reducew1,r,T
See Also
Basis
MonomialOrder
Reduce
SPolynomial
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