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Macaulay2Doc :: minimalPresentation(Ideal)

minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal

Synopsis

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];
i2 : I = ideal(x-x^2-y,z+x*y,w^2-u^2);

o2 : Ideal of C
i3 : minPres I

              2    2
o3 = ideal(- u  + w )

               ZZ
o3 : Ideal of ---[x, u, w]
              101
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
i5 : I.cache.minimalPresentationMapInv

              ZZ
o5 = map (C, ---[x, u, w], {x, u, w})
             101

                     ZZ
o5 : RingMap C <--- ---[x, u, w]
                    101
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];
i7 : I = ideal(x-x^2-y,z+x*y,w^2-u^2);

o7 : Ideal of R
i8 : minimalPresentation(I, Exclude=>{1})

               2             2    2
o8 = ideal (- x  + x - y, - u  + w )

               ZZ
o8 : Ideal of ---[x..y, u, w]
              101

See also