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Groebner[HilbertDimension] - compute Hilbert Dimension

Groebner[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.

withGroebner:

F:=x22xz+5,xy2+yz3,3y28z3

F:=x22xz+5,yz3+xy2,8z3+3y2

(1)

HilbertDimensionF

0

(2)

mapUnivariatePolynomial,x,y,z,F

3x1264x11+90x10960x9+1125x84800x7+7500x68000x5+28125x4+56250x2+46875,729y8+41472y7+77760y6+2764800y4+32768000y2,9z996z8+240z6+1600z3

(3)

MaximalIndependentSetF

(4)

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

HilbertDimensionF1..2

1

(5)

MaximalIndependentSetF1..2

y

(6)

mapUnivariatePolynomial,x,y,z,F1..2

0,0,0

(7)

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

HilbertDimensionF,characteristic=2

1

(8)

mapUnivariatePolynomial,x,y,z,F,characteristic=2

x2+1,y2,0

(9)

MaximalIndependentSetF,characteristic=2

z

(10)

See Also

Basis, HilbertSeries, IsZeroDimensional, PolynomialIdeals[HilbertDimension], UnivariatePolynomial


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