The MaximalIndependentSet command computes a maximal set of (algebraically) independent variables U such that the intersection of J with the subring K[U] is empty.  The number of elements in such a set is equal to the Hilbert dimension of the ideal, as well as the affine dimension of the corresponding variety.

In the case of skew polynomials, the dimension that is returned is that of the left ideal generated by J.

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 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 HilbertDimension and MaximalIndependentSet uses the leading monomials of a total degree Groebner basis for J. To access this functionality directly (as part of a program), make J the list or set of leading monomials. The commands will detect this case and execute the algorithm with minimal overhead.

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

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

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

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

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

$0$

$\mathrm{map}\left(\mathrm{UnivariatePolynomial}\,\left[x\,y\,z\right]\,F\right)$

$\left[3x^{12}-64x^{11}+90x^{10}-960x^9+1125x^8-4800x^7+7500x^6-8000x^5+28125x^4+56250x^2+46875\,729y^8+41472y^7+77760y^6+2764800y^4+32768000y^2\,9z^9-96z^8+240z^6+1600z^3\right]$

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

$\varnothing$

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

$1$

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

$\left\{y\right\}$

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

$\left[0\,0\,0\right]$

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

$1$

$\mathrm{map}\left(\mathrm{UnivariatePolynomial}\,\left[x\,y\,z\right]\,F\,\mathrm{characteristic}=2\right)$

$\left[x^2+1\,y^2\,0\right]$

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

$\left\{z\right\}$