Groebner - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Algebra : Polynomials : Groebner : Groebner/IsBasis

Groebner

  

IsBasis

  

test for a Groebner basis

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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:

Gx2+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)

sSPolynomialG1,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.

HBasisG&comma;plexx&comma;y

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

(6)

IsBasisH&comma;plexx&comma;y

true

(7)

Compatibility

• 

The Groebner[IsBasis] command was introduced in Maple 16.

• 

For more information on Maple 16 changes, see Updates in Maple 16.

See Also

Basis

Monomial Orders

MonomialOrder

NormalForm

SPolynomial

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam