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PolynomialIdeals

  

VanishingIdeal

  

compute the vanishing ideal for finite a set of points

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

VanishingIdeal(S, X)

VanishingIdeal(S, X, T, p)

Parameters

S

-

list or set of points

X

-

list of variable names

T

-

(optional) monomial order

p

-

(optional) characteristic, a non-negative integer

Description

• 

The VanishingIdeal command constructs the vanishing ideal for a set of points in affine space.  The output of this command is the ideal of polynomials that vanish (that is, are identically zero) on S.

• 

The first argument must be a list or set of points in affine space. Each point is given as a list with coordinates corresponding to the variables in X.

• 

The third argument is optional, and specifies a monomial order for which a Groebner basis is computed. If omitted, VanishingIdeal chooses lexicographic order, which is generally the fastest order.

• 

The field characteristic can be specified with an optional last argument.  The default is characteristic zero.

• 

Multiple occurrences of the same point in S are ignored, so that VanishingIdeal always returns a radical ideal.

Examples

withPolynomialIdeals:

L5,4,4,4,0,2,6,4,1,3,0,5,3,1,3

L5,4,4,4,0,2,6,4,1,3,0,5,3,1,3

(1)

JVanishingIdealL,x,y,z

Jz515z4+85z3225z2+274z120,9z498z3+351z2+24x454z+48,z48z3+13z2+4y+18z40

(2)

SimplifyPrimeDecompositionJ

y,3+x,z5,5+x,4+y,z4,3+x,1+y,z3,y,4+x,z2,6+x,4+y,z1

(3)

VanishingIdealL,x,y,z,tdegx,y,z

13y236x37y12z+168,13yz+26z248x19y198z+484,78xz+91z2354x+50y937z+2302,39xy26z236x193y+170z92,39x226z2255x40y+188z+124,13z3117z212x+18y+308z204

(4)

aliasα=RootOfz3+z+1

α

(5)

M1,α,α2α1,0,1,1α

M1,α,α2α1,0,1,1α

(6)

KVanishingIdealM,x,y,2

Kα2y2+α2+x+y+1,α2y2+y2α+y3+y2+y

(7)

IdealInfo[Characteristic]K

2

(8)

SimplifyPrimeDecompositionK

y,α2+x+1,x+1,y+α,x+1,α2+y+1

(9)

References

  

Farr, Jeff. Computing Grobner bases, with applications to Pade approximation and algebraic coding theory. Ph.D. Thesis, Clemson University, 2003.

See Also

alias

Groebner[Basis]

MonomialOrders

PolynomialIdeals

PolynomialIdeals[IdealInfo]

PolynomialIdeals[PrimeDecomposition]

PolynomialIdeals[Simplify]

 


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