HilbertDimension - Maple Help

Groebner

 HilbertDimension
 compute Hilbert Dimension
 MaximalIndependentSet
 compute maximal independent set

 Calling Sequence HilbertDimension(J, X, characteristic=p) MaximalIndependentSet(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 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.

Examples

The ideal below is zero-dimensional, so the set of solutions are points in C[x,y,z]. The intersection of F with each variable is a univariate polynomial so there are no algebraically independent variables.

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[{x}^{2}-2xz+5,x{y}^{2}+y{z}^{3},3{y}^{2}-8{z}^{3}\right]$
 ${F}{≔}\left[{{x}}^{{2}}{-}{2}{}{x}{}{z}{+}{5}{,}{y}{}{{z}}^{{3}}{+}{x}{}{{y}}^{{2}}{,}{-}{8}{}{{z}}^{{3}}{+}{3}{}{{y}}^{{2}}\right]$ (1)
 > $\mathrm{HilbertDimension}\left(F\right)$
 ${0}$ (2)
 > $\mathrm{map}\left(\mathrm{UnivariatePolynomial},\left[x,y,z\right],F\right)$
 $\left[{3}{}{{x}}^{{12}}{-}{64}{}{{x}}^{{11}}{+}{90}{}{{x}}^{{10}}{-}{960}{}{{x}}^{{9}}{+}{1125}{}{{x}}^{{8}}{-}{4800}{}{{x}}^{{7}}{+}{7500}{}{{x}}^{{6}}{-}{8000}{}{{x}}^{{5}}{+}{28125}{}{{x}}^{{4}}{+}{56250}{}{{x}}^{{2}}{+}{46875}{,}{729}{}{{y}}^{{8}}{+}{41472}{}{{y}}^{{7}}{+}{77760}{}{{y}}^{{6}}{+}{2764800}{}{{y}}^{{4}}{+}{32768000}{}{{y}}^{{2}}{,}{9}{}{{z}}^{{9}}{-}{96}{}{{z}}^{{8}}{+}{240}{}{{z}}^{{6}}{+}{1600}{}{{z}}^{{3}}\right]$ (3)
 > $\mathrm{MaximalIndependentSet}\left(F\right)$
 ${\varnothing }$ (4)

The first two equations generate a curve in C[x,y,z]. All of the variables are algebraically independent.

 > $\mathrm{HilbertDimension}\left(F\left[1..2\right]\right)$
 ${1}$ (5)
 > $\mathrm{MaximalIndependentSet}\left(F\left[1..2\right]\right)$
 $\left\{{y}\right\}$ (6)
 > $\mathrm{map}\left(\mathrm{UnivariatePolynomial},\left[x,y,z\right],F\left[1..2\right]\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (7)

Over GF(2) the situation is different, z is algebraically independent so the ideal generates a "curve".

 > $\mathrm{HilbertDimension}\left(F,\mathrm{characteristic}=2\right)$
 ${1}$ (8)
 > $\mathrm{map}\left(\mathrm{UnivariatePolynomial},\left[x,y,z\right],F,\mathrm{characteristic}=2\right)$
 $\left[{{x}}^{{2}}{+}{1}{,}{{y}}^{{2}}{,}{0}\right]$ (9)
 > $\mathrm{MaximalIndependentSet}\left(F,\mathrm{characteristic}=2\right)$
 $\left\{{z}\right\}$ (10)