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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

Parameters

Description

Examples

References

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 kX.  This set has the property that J∩kX=0.  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

withPolynomialIdeals:

Jx2y2+z4w,xzyw2

J:=w2y+xz,wz4+x2y2

(1)

J is in Q[w, x, y, z].

IsZeroDimensionalJ

false

(2)

MMaximalIndependentSetJ

M:=y,z

(3)

EliminationIdealJ,M

0

(4)

HilbertDimensionJ

2

(5)

VIdealInfo:-VariablesJ

V:=w,x,y,z

(6)

HilbertDimensionJ,V∖M

0

(7)

References

  

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

See Also

Groebner[Basis]

Groebner[HilbertDimension]

Groebner[IsZeroDimensional]

MonomialOrders

PolynomialIdeals

PolynomialIdeals[EliminationIdeal]

PolynomialIdeals[IdealInfo]

 


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