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