Description
If the ideal
I is homogeneous, then the ideal
J, in a new ring
Q is the defining ideal for a minimal presentation of the ring
S/I where
S is the ring of
I. This is accomplished as follows. If a variable occurs as a term of a generator of
I and in no other terms of the same polynomial, then the variable is replaced by the remaining terms and removed from the ring. A minimal generating set for the resulting ideal is then computed. If
I is not homogeneous, then an attempt is made to improve the presentation of
S/I.
i1 : C = ZZ/101[x,y,z,u,w];
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i2 : I = ideal(x-x^2-y,z+x*y,w^2-u^2);
o2 : Ideal of C
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i3 : minPres I
2 2
o3 = ideal(- u + w )
ZZ
o3 : Ideal of ---[x, u, w]
101
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i4 : I.cache.minimalPresentationMap
ZZ 2 3 2
o4 = map (---[x, u, w], C, {x, - x + x, x - x , u, w})
101
ZZ
o4 : RingMap ---[x, u, w] <-- C
101
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i5 : I.cache.minimalPresentationMapInv
ZZ
o5 = map (C, ---[x, u, w], {x, u, w})
101
ZZ
o5 : RingMap C <-- ---[x, u, w]
101
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If the Exclude option is present, then those variables with the given indices are not simplified away (remember that ring variable indices start at 0).
i6 : R = ZZ/101[x,y,z,u,w];
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i7 : I = ideal(x-x^2-y,z+x*y,w^2-u^2);
o7 : Ideal of R
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i8 : minimalPresentation(I, Exclude=>{1})
2 2 2
o8 = ideal (- x + x - y, - u + w )
ZZ
o8 : Ideal of ---[x..y, u, w]
101
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