PolynomialIdeals
Contract
contract an ideal to a larger ring
Calling Sequence
Parameters
Description
Examples
References
Contract(J, X)
J
-
polynomial ideal
X
set of variable names
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.
withPolynomialIdeals:
Jxy≔z2x2−y,z3xy,variables=x,y
Jxy≔z3xy,z2x2−y
J≔ContractJxy,x,y,z
J≔x3,y2,xy,z2x2−y
IdealInfo:-VariablesJ
x,y,z
K≔IntersectJ,y2−xz
K≔−xyz+y3,−x3z+x2y2
zdd≔ZeroDimensionalDecompositionK
zdd≔−xz+y2,x2,−xyz+y3
seqHilbertDimensioni,i=zdd
0,0
seqIdealInfo:-Variablesi,i=zdd
y,x,y
C≔seqContracti,x,y,z,i=zdd
C≔−xz+y2,x2,−xyz+y3
seqIdealInfo:-Variablesi,i=C
x,y,z,x,y,z
L≔IntersectC
L≔xyz−y3,x3z−x2y2
IdealContainmentK,L,K
true
Becker, T., and Weispfenning, V. Groebner Bases. New York: Springer-Verlag, 1993.
See Also
PolynomialIdeals[HilbertDimension]
PolynomialIdeals[IdealContainment]
PolynomialIdeals[IdealInfo]
PolynomialIdeals[Intersect]
PolynomialIdeals[ZeroDimensionalDecomposition]
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