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Macaulay2Doc :: minimalPresentation(Ring)

minimalPresentation(Ring) -- compute a minimal presentation of a quotient ring

Synopsis

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);
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(x-x^2-y,z+x*y,w^2-u^2);
i6 : minimalPresentation(R, Exclude=>{1})

           ZZ
          ---[x, y, u, w]
          101
o6 = -------------------------
         2             2    2
     (- x  + x - y, - u  + w )

o6 : QuotientRing

See also