test for a Groebner basis
IsBasis(G, T, characteristic=p)
set or list of polynomials
MonomialOrder or ShortMonomialOrder
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.
G ≔ x2+1,y2+x+1
Our example shows that whether G is not a Groebner basis or not depends on the monomial ordering.
s ≔ SPolynomial⁡G1,G2,plex⁡x,y
Now we compute a (reduced) Groebner basis for the ideal generated by G in the lexicographical monomial ordering with y<x.
H ≔ Basis⁡G,plex⁡x,y
The Groebner[IsBasis] command was introduced in Maple 16.
For more information on Maple 16 changes, see Updates in Maple 16.
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