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.

$\mathrm{with}\left(\mathrm{Groebner}\right)\:$

$G≔\left[x^2+1\,y^2+x+1\right]$

$G≔\left[x^2+1\,y^2+x+1\right]$

$\mathrm{IsBasis}\left(G\,\mathrm{grlex}\left(x\,y\right)\right)$

$\mathrm{true}$

$\mathrm{IsBasis}\left(G\,\mathrm{plex}\left(x\,y\right)\right)$

$\mathrm{false}$

$s≔\mathrm{SPolynomial}\left(G\left[1\right]\,G\left[2\right]\,\mathrm{plex}\left(x\,y\right)\right)$

$s≔-x{y}^2-x+1$

$\mathrm{NormalForm}\left(s\,G\,\mathrm{plex}\left(x\,y\right)\right)$

$y^4+2y^2+2$

$H≔\mathrm{Basis}\left(G\,\mathrm{plex}\left(x\,y\right)\right)$

$H≔\left[y^4+2y^2+2\,y^2+x+1\right]$

$\mathrm{IsBasis}\left(H\,\mathrm{plex}\left(x\,y\right)\right)$

$\mathrm{true}$

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

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