 type/ClosedIdeal - Maple Programming Help

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type/ClosedIdeal

type for finite-dimensional ideals

 Calling Sequence type(G, ClosedIdeal(T))

Parameters

 G - set or list of polynomials T - table that denotes a monomial ordering on an algebra

Description

 • 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).

Examples

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $A≔\mathrm{poly_algebra}\left(x,y,z\right):$
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(x,y,z\right)\right):$
 > $F≔\left[{x}^{2}-2xz+5,x{y}^{2}+y{z}^{3},3{y}^{2}-8{z}^{3}\right]:$
 > $\mathrm{type}\left(F,\mathrm{ClosedIdeal}\left(T\right)\right)$
 ${\mathrm{false}}$ (1)

Thus far, no Groebner basis has been computed.

 > $G≔\mathrm{Basis}\left(F,T\right):$
 > $\mathrm{type}\left(G,\mathrm{ClosedIdeal}\left(T\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsZeroDimensional}\left(F\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsZeroDimensional}\left(G\right)$
 ${\mathrm{true}}$ (4)