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(xx^2y,z+x*y,w^2u^2);

i2 : minimalPresentation(R)
ZZ
[x, u, w]
101
o2 = 
2 2
 u + w
o2 : QuotientRing

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

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

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(xx^2y,z+x*y,w^2u^2);

i6 : minimalPresentation(R, Exclude=>{1})
ZZ
[x, y, u, w]
101
o6 = 
2 2 2
( x + x  y,  u + w )
o6 : QuotientRing
