The type ClosedIdeal checks if the leading monomials of G with respect to T generate a zero-dimensional ideal.

When G is a Groebner basis with respect to T, the call type(G, T) is equivalent to the call Groebner[IsZeroDimensional](G, T), and checks if the ideal generated by G is finite-dimensional. type/ClosedIdeal is therefore less general but does not compute any Groebner basis (as opposed to IsZeroDimensional).

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\:$

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

$A≔\mathrm{poly\_algebra}\left(x\,y\,z\right)\:$

$T≔\mathrm{MonomialOrder}\left(A\,\mathrm{tdeg}\left(x\,y\,z\right)\right)\:$

$F≔\left[x^2-2xz+5\,x{y}^2+y{z}^3\,3y^2-8z^3\right]\:$

$\mathrm{type}\left(F\,\mathrm{ClosedIdeal}\left(T\right)\right)$

$\mathrm{false}$

$G≔\mathrm{Basis}\left(F\,T\right)\:$

$\mathrm{type}\left(G\,\mathrm{ClosedIdeal}\left(T\right)\right)$

$\mathrm{true}$

$\mathrm{IsZeroDimensional}\left(F\right)$

$\mathrm{true}$

$\mathrm{IsZeroDimensional}\left(G\right)$

$\mathrm{true}$