VanishingIdeal - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


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,x3,z5,5+x,4+y,z4,1+y,x3,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+2302,39yx26z236x193y+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)

IdealInfoCharacteristicK

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]