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PolynomialIdeals[PrimaryDecomposition] - compute a primary decomposition of an ideal

PolynomialIdeals[PrimeDecomposition] - compute a prime decomposition of the radical of an ideal

Calling Sequence

PrimaryDecomposition(J, k)

PrimeDecomposition(J, k)

Parameters

J

-

polynomial ideal

k

-

(optional) field extension

Description

• 

The PrimaryDecomposition command constructs a finite sequence of primary ideals whose intersection equals the input J.  Likewise the PrimeDecomposition command constructs a sequence of prime ideals whose intersection is equal to the radical of J. Calling PrimeDecomposition(J) is faster but otherwise equivalent to calling PrimaryDecomposition(Radical(J)).

• 

By default, ideals are factored over the domain implied by their coefficients - usually the rationals or the integers mod p.  Additional field extensions can be specified with an optional second argument k, which can be a single RootOf or radical, or a list or set of RootOfs and radicals.

• 

The output of these commands is not canonical, and may not be unique. However, a Groebner basis is stored for each ideal in the sequence so the Simplify command can be used at no additional cost.

• 

The algorithms employed by these commands require polynomials over a perfect field.  Infinite fields of positive characteristic are not supported. Over finite fields, only zero-dimensional ideals can be handled because the dimension reducing process generates infinite fields.

Examples

withPolynomialIdeals:

J:=x2,xy+x

J:=x2,xy+x

(1)

PrimeDecompositionJ

x

(2)

PrimaryDecompositionJ

x,x2,y+1

(3)

J:=x22,y2+1,z2+2

J:=x22,y2+1,z2+2

(4)

P:=PrimaryDecompositionJ

P:=x22,y2+1,z2+2,xy+z,x22,y2+1,z2+2,xy+z

(5)

IntersectP

x22,y2+1,z2+2

(6)

K:=J,characteristic=5

K:=x2+3,y2+1,z2+2

(7)

P:=PrimaryDecompositionK

P:=y+2,z+2x,x2+3,y2+1,z2+2,y+2,z+3x,x2+3,y2+1,z2+2,y+3,z+2x,x2+3,y2+1,z2+2,y+3,z+3x,x2+3,y2+1,z2+2

(8)

P:=SimplifyP

P:=y+2,z+2x,x2+3,y+2,z+3x,x2+3,y+3,z+2x,x2+3,y+3,z+3x,x2+3

(9)

IntersectP

x2+3,y2+1,z2+2

(10)

L:=xz22xy,y2z2x

L:=xz2+y2,xz22xy

(11)

P:=PrimaryDecompositionL

P:=x,y2,y,z2,z2+4x,z2+2y

(12)

IdealContainmentL,IntersectP,L

true

(13)

H:=x2+y21,x+y

H:=x+y,x2+y21

(14)

PrimeDecompositionH

x+y,x2+y21

(15)
k=Qalpha=Q2

aliasα=RootOfz22

α

(16)

P:=PrimeDecompositionH,α

P:=x+y,2xα,x2+y21,x+y,2x+α,x2+y21

(17)

SimplifyP

2xα,α+2y,2x+α,α+2y

(18)

See Also

Groebner[Solve], PolynomialIdeals, PolynomialIdeals[IdealContainment], PolynomialIdeals[Intersect], PolynomialIdeals[IsPrimary], PolynomialIdeals[IsPrime], PolynomialIdeals[Radical], PolynomialIdeals[Simplify], PolynomialIdeals[ZeroDimensionalDecomposition]

References

  

Gianni, P.; Trager, B.; and Zacharias, G. "Grobner bases and primary decompositions of polynomial ideals." J. Symbolic Comput. Vol. 6, (1988): 149-167.


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