The IsZeroDimensional command decides whether a set of polynomials J with respect to the indeterminates X has a finite number of solutions over the algebraic closure of the coefficient field.  For example, in characteristic zero this command tests whether there are a finite number of solutions in the complex numbers. In every domain this test is equivalent to testing whether the HilbertDimension is zero.

The variables of the system can be specified using an optional second argument X. If X is a ShortMonomialOrder then a Groebner basis of J with respect to X is computed. By default, X is the set of all indeterminates not appearing inside a RootOf command or radical when J is a list or set, or PolynomialIdeals[IdealInfo][Variables](J) if J is an ideal.

The optional argument characteristic=p specifies the ring characteristic when J is a list or set. This option has no effect when J is a PolynomialIdeal or when X is a MonomialOrder.

The algorithm for IsZeroDimensional tests whether a power of each variable appears as a leading monomial in a Groebner basis for J. To access this functionality directly (as a subroutine in your program), make J a list or set of leading monomials. IsZeroDimensional will detect this case and execute the algorithm with minimal overhead.

Note that the is_finite command is deprecated.  It may not be supported in a future Maple release.

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

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

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

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

$\mathrm{true}$

$\mathrm{LeadingMonomial}\left(\mathrm{Basis}\left(F\,\mathrm{tdeg}\left(x\,y\,z\right)\right)\,\mathrm{tdeg}\left(x\,y\,z\right)\right)$

$\left[x^2\,z^3\,x{y}^2\,y^4\right]$

$\mathrm{IsZeroDimensional}\left(F\,\mathrm{characteristic}=2\right)$

$\mathrm{false}$

$\mathrm{LeadingMonomial}\left(\mathrm{Basis}\left(F\,\mathrm{tdeg}\left(x\,y\,z\right)\,\mathrm{characteristic}=2\right)\,\mathrm{tdeg}\left(x\,y\,z\right)\right)$

$\left[y^2\,x^2\,y{z}^3\right]$

$\mathrm{IsZeroDimensional}\left(F\left[1..2\right]\right)$

$\mathrm{false}$

$\mathrm{HilbertDimension}\left(F\left[1..2\right]\right)$

$1$

$\mathrm{IsZeroDimensional}\left(F\left[1..2\right]\,\left\{x\,y\right\}\right)$

$\mathrm{true}$

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

$J≔⟨F⟩$

$J≔⟨-8z^3+3y^2\,y{z}^3+x{y}^2\,x^2-2xz+5⟩$

$\mathrm{NumberOfSolutions}\left(J\right)$

$18$

$\mathrm{NormalSet}\left(J\,\mathrm{tdeg}\left(x\,y\,z\right)\right)\left[1\right]$

$\left[1\,z\,y\,x\,z^2\,yz\,xz\,y^2\,xy\,y{z}^2\,x{z}^2\,y^{2}z\,xyz\,y^3\,y^2{z}^2\,xy{z}^2\,y^{3}z\,y^3{z}^2\right]$