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Groebner

 IsZeroDimensional
 decide if a system has a finite number of solutions

 Calling Sequence IsZeroDimensional(J, X, characteristic=p)

Parameters

 J - a list or set of polynomials or a PolynomialIdeal X - (optional) a list or set of variables, a ShortMonomialOrder, or a MonomialOrder p - (optional) characteristic

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[{x}^{2}-2xz+5,x{y}^{2}+y{z}^{3},3{y}^{2}-8{z}^{3}\right]$
 ${F}{:=}\left[{{x}}^{{2}}{-}{2}{}{x}{}{z}{+}{5}{,}{y}{}{{z}}^{{3}}{+}{x}{}{{y}}^{{2}}{,}{-}{8}{}{{z}}^{{3}}{+}{3}{}{{y}}^{{2}}\right]$ (1)
 > $\mathrm{IsZeroDimensional}\left(F\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{LeadingMonomial}\left(\mathrm{Basis}\left(F,\mathrm{tdeg}\left(x,y,z\right)\right),\mathrm{tdeg}\left(x,y,z\right)\right)$
 $\left[{{x}}^{{2}}{,}{{z}}^{{3}}{,}{x}{}{{y}}^{{2}}{,}{{y}}^{{4}}\right]$ (3)
 > $\mathrm{IsZeroDimensional}\left(F,\mathrm{characteristic}=2\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{LeadingMonomial}\left(\mathrm{Basis}\left(F,\mathrm{tdeg}\left(x,y,z\right),\mathrm{characteristic}=2\right),\mathrm{tdeg}\left(x,y,z\right)\right)$
 $\left[{{y}}^{{2}}{,}{{x}}^{{2}}{,}{y}{}{{z}}^{{3}}\right]$ (5)
 > $\mathrm{IsZeroDimensional}\left({F}_{1..2}\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{HilbertDimension}\left({F}_{1..2}\right)$
 ${1}$ (7)
 > $\mathrm{IsZeroDimensional}\left({F}_{1..2},\left\{x,y\right\}\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨F⟩$
 ${J}{:=}⟨{-}{8}{}{{z}}^{{3}}{+}{3}{}{{y}}^{{2}}{,}{y}{}{{z}}^{{3}}{+}{x}{}{{y}}^{{2}}{,}{{x}}^{{2}}{-}{2}{}{x}{}{z}{+}{5}⟩$ (9)
 > $\mathrm{NumberOfSolutions}\left(J\right)$
 ${18}$ (10)
 > ${\mathrm{NormalSet}\left(J,\mathrm{tdeg}\left(x,y,z\right)\right)}_{1}$
 $\left[{1}{,}{z}{,}{y}{,}{x}{,}{{z}}^{{2}}{,}{y}{}{z}{,}{x}{}{z}{,}{{y}}^{{2}}{,}{x}{}{y}{,}{y}{}{{z}}^{{2}}{,}{x}{}{{z}}^{{2}}{,}{{y}}^{{2}}{}{z}{,}{x}{}{y}{}{z}{,}{{y}}^{{3}}{,}{{y}}^{{2}}{}{{z}}^{{2}}{,}{x}{}{y}{}{{z}}^{{2}}{,}{{y}}^{{3}}{}{z}{,}{{y}}^{{3}}{}{{z}}^{{2}}\right]$ (11)