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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, MonomialOrder => GRevLex => 3]
              101
i4 : I.cache.minimalPresentationMap

          ZZ                                                   2       3  
o4 = map(--- [x, u, w, MonomialOrder => GRevLex => 3],C,{x, - x  + x, x  -
         101
     ------------------------------------------------------------------------
      2
     x , u, w})

              ZZ
o4 : RingMap --- [x, u, w, MonomialOrder => GRevLex => 3] <--- C
             101
i5 : I.cache.minimalPresentationMapInv

            ZZ
o5 = map(C,--- [x, u, w, MonomialOrder => GRevLex => 3],{x, u, w})
           101

                     ZZ
o5 : RingMap C <--- --- [x, u, w, MonomialOrder => GRevLex => 3]
                    101

See also