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Groebner

  

InterReduce

  

inter-reduce a list of polynomials

 

Calling Sequence

Parameters

Description

Examples

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

withGroebner:

Fx2+xy2,x2xy

F:=x2+xy2,x2xy

(1)

LeadingMonomialF,tdegx,y

x2,x2

(2)

InterReduceF,tdegx,y

xy1,x21

(3)

rReduceF1,F2,tdegx,y

r:=xy1

(4)

ReduceF2,r,tdegx,y

x21

(5)

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

SPolynomialxy1,x21,tdegx,y

x+y

(6)

Groebner[Basis]F,tdegx,y

xy,y21

(7)

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

withOre_algebra:

Adiff_algebraDn,n

A:=Ore_algebra

(8)

TMonomialOrderA,tdegDn

T:=monomial_order

(9)

w1n2Dn2+n

w1:=Dn2n2+n

(10)

w2n2Dn2+Dn

w2:=Dn2n2+Dn

(11)

InterReducew1,w2,T

1

(12)

rReducew2,w1,T

r:=Dnn

(13)

Reducew1,r,T

1

(14)

See Also

Basis

MonomialOrder

Reduce

SPolynomial

 


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