Description
The computation is accomplished by considering the relations of
R. If a variable occurs as a term of a relation of
R 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 defining ideal is then computed and the new quotient ring is returned. If
R is not homogeneous, then an attempt is made to improve the presentation.
i1 : R = ZZ/101[x,y,z,u,w]/ideal(x-x^2-y,z+x*y,w^2-u^2);
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i2 : minimalPresentation(R)
ZZ
---[x, u, w]
101
o2 = ------------
2 2
- u + w
o2 : QuotientRing
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i3 : R.minimalPresentationMap
ZZ
---[x, u, w]
101 2 3 2
o3 = map (------------, R, {x, - x + x, x - x , u, w})
2 2
- u + w
ZZ
---[x, u, w]
101
o3 : RingMap ------------ <-- R
2 2
- u + w
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i4 : R.minimalPresentationMapInv
ZZ
---[x, u, w]
101
o4 = map (R, ------------, {x, u, w})
2 2
- u + w
ZZ
---[x, u, w]
101
o4 : RingMap R <-- ------------
2 2
- u + w
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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).
i5 : R = ZZ/101[x,y,z,u,w]/ideal(x-x^2-y,z+x*y,w^2-u^2);
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i6 : minimalPresentation(R, Exclude=>{1})
ZZ
---[x..y, u, w]
101
o6 = -------------------------
2 2 2
(- x + x - y, - u + w )
o6 : QuotientRing
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