HilbertDimension - Maple Help

PolynomialIdeals

 HilbertDimension
 compute the Hilbert dimension of an ideal
 MaximalIndependentSet
 compute a maximal independent set of variables
 IsZeroDimensional
 test if an ideal is zero-dimensional

 Calling Sequence HilbertDimension(J, X) MaximalIndependentSet(J, X) IsZeroDimensional(J, X)

Parameters

 J - polynomial ideal X - (optional) set of ring variable names

Description

 • The HilbertDimension command computes the Hilbert dimension of an ideal.
 • The MaximalIndependentSet command computes a maximal independent set of variables for an ideal J in ${k}_{X}$.  This set has the property that $J\cap {k}_{X}=\left\{0\right\}$.  The cardinality of this set is an invariant, equal to the Hilbert dimension of the ideal. These commands require a total degree Groebner basis.
 • The IsZeroDimensional command tests only whether an ideal has Hilbert dimension zero.  This can be done using any Groebner basis. In cases where the dimension is not zero, some computation is avoided.
 • An optional second argument can be used to override the variables of the polynomial ring.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{x}^{2}-{y}^{2}+{z}^{4}w,xz-y{w}^{2}⟩$
 ${J}{≔}⟨{-}{y}{}{{w}}^{{2}}{+}{x}{}{z}{,}{{z}}^{{4}}{}{w}{+}{{x}}^{{2}}{-}{{y}}^{{2}}⟩$ (1)

J is in Q[w, x, y, z].

 > $\mathrm{IsZeroDimensional}\left(J\right)$
 ${\mathrm{false}}$ (2)
 > $M≔\mathrm{MaximalIndependentSet}\left(J\right)$
 ${M}{≔}\left\{{y}{,}{z}\right\}$ (3)
 > $\mathrm{EliminationIdeal}\left(J,M\right)$
 $⟨{0}⟩$ (4)
 > $\mathrm{HilbertDimension}\left(J\right)$
 ${2}$ (5)
 > $V≔\mathrm{IdealInfo}:-\mathrm{Variables}\left(J\right)$
 ${V}{≔}\left\{{w}{,}{x}{,}{y}{,}{z}\right\}$ (6)
 > $\mathrm{HilbertDimension}\left(J,V\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{minus}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}M\right)$
 ${0}$ (7)

References

 Becker, T., and Weispfenning, V. Groebner Bases. New York: Springer-Verlag, 1993.