next | previous | forward | backward | up | top | index | toc | Macaulay2 website
Macaulay2Doc > basic commutative algebra > M2SingularBook > Singular Book 1.6.13

Singular Book 1.6.13 -- normal form

Normal forms in Macaulay2 are done using the remainder operator %.
i1 : R = QQ[x,y,z];
i2 : f = x^2*y*z+x*y^2*z+y^2*z+z^3+x*y;
i3 : f1 = x*y+y^2-1

            2
o3 = x*y + y  - 1

o3 : R
i4 : f2 = x*y

o4 = x*y

o4 : R
i5 : G = ideal(f1,f2)

                   2
o5 = ideal (x*y + y  - 1, x*y)

o5 : Ideal of R
Macaulay2 computes a Groebner basis of G, and uses that to find the normal form of f. In Macaulay2, all remainders are reduced normal forms (at least for non-local orders).
i6 : f % G

      3
o6 = z  + z

o6 : R

In order to reduce using a non Groebner basis, use forceGB

i7 : f % (forceGB gens G)

      2     3    2
o7 = y z + z  - y  + x*z + 1

o7 : R
This is a different answer from the SINGULAR book, since the choice of divisor affects the answer.
i8 : f % (forceGB matrix{{f2,f1}})

      2     3
o8 = y z + z

o8 : R