test for a Groebner basis - Maple Help

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Groebner[IsBasis] - test for a Groebner basis

Calling Sequence

IsBasis(G, T)

IsBasis(G, T, characteristic=p)

Parameters

G

-

set or list of polynomials

T

-

MonomialOrder or ShortMonomialOrder

p

-

(optional) characteristic

Description

• 

IsBasis(G, T) outputs true if G is a Groebner basis for the ideal I generated by G with respect to the monomial order T and false otherwise.

• 

The test applies Buchberger's S-polynomial criterion which states that G is a Groebner basis for I if and only if the S-polynomial of each pair of polynomials in G when divided by G has 0 remainder.  Note, this test can take longer than the time it takes to compute the Groebner basis.

• 

The argument T is a monomial order.  For a list of available monomial orders, see the Monomial Orders help page.

• 

An optional argument characteristic=p can be used to specify the ring characteristic. The default value is zero.

Examples

withGroebner:

G:=x2+1,y2+x+1

G:=x2+1,y2+x+1

(1)

Our example shows that whether G is not a Groebner basis or not depends on the monomial ordering.

IsBasisG,grlexx,y

true

(2)

IsBasisG,plexx,y

false

(3)

s:=SPolynomialG1,G2,plexx,y

s:=xy2x+1

(4)

NormalForms,G,plexx,y

y4+2y2+2

(5)

Now we compute a (reduced) Groebner basis for the ideal generated by G in the lexicographical monomial ordering with y<x.

H:=BasisG&comma;plexx&comma;y

H:=y4&plus;2y2&plus;2&comma;y2&plus;x&plus;1

(6)

IsBasisH&comma;plexx&comma;y

true

(7)

See Also

Basis, Monomial Orders, MonomialOrder, NormalForm, SPolynomial


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