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PolynomialIdeals[Contract] - contract an ideal to a larger ring

Calling Sequence

Contract(J, X)

Parameters

J

-

polynomial ideal

X

-

set of variable names

Description

• 

The Contract command contracts the image of an ideal in kUXU back to the polynomial ring kX.

• 

The typical use of this command is to contract the result of a zero-dimensional decomposition back to the original ring. This is part of a process used to extend algorithms for zero-dimensional ideals to ideals of positive dimension. For additional details, see the help page for ZeroDimensionalDecomposition.

Examples

withPolynomialIdeals:

Jxy:=z2x2y,z3xy,variables=x,y

Jxy:=z3xy,x2z2y

(1)

J:=ContractJxy,x,y,z

J:=x3,y2,xy,x2z2y

(2)

IdealInfo:-VariablesJ

x,y,z

(3)

K:=IntersectJ,y2xz

K:=xyz+y3,x3z+x2y2

(4)

zdd:=ZeroDimensionalDecompositionK

zdd:=xz+y2,x2,xyz+y3

(5)

seqHilbertDimensioni,i=zdd

0,0

(6)

seqIdealInfo:-Variablesi,i=zdd

y,x,y

(7)

C:=seqContracti,x,y,z,i=zdd

C:=xz+y2,x2,xyz+y3

(8)

seqIdealInfo:-Variablesi,i=C

x,y,z,x,y,z

(9)

L:=IntersectC

L:=xyzy3,x3zx2y2

(10)

IdealContainmentK,L,K

true

(11)

See Also

PolynomialIdeals, PolynomialIdeals[HilbertDimension], PolynomialIdeals[IdealContainment], PolynomialIdeals[IdealInfo], PolynomialIdeals[Intersect], PolynomialIdeals[ZeroDimensionalDecomposition]

References

  

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


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