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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp1.6.13.html" title="normal form">Singular Book 1.6.13</a></div>
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<div><h1>Singular Book 1.6.13 -- normal form</h1>
<div>Normal forms in Macaulay2 are done using the remainder operator <a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : f = x^2*y*z+x*y^2*z+y^2*z+z^3+x*y;</pre>
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<tr><td><pre>i3 : f1 = x*y+y^2-1
2
o3 = x*y + y - 1
o3 : R</pre>
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<tr><td><pre>i4 : f2 = x*y
o4 = x*y
o4 : R</pre>
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<tr><td><pre>i5 : G = ideal(f1,f2)
2
o5 = ideal (x*y + y - 1, x*y)
o5 : Ideal of R</pre>
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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).<table class="examples"><tr><td><pre>i6 : f % G
3
o6 = z + z
o6 : R</pre>
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In order to reduce using a non Groebner basis, use <a href="_force__G__B.html" title="declare that the columns of a matrix are a Gröbner basis">forceGB</a><table class="examples"><tr><td><pre>i7 : f % (forceGB gens G)
2 3 2
o7 = y z + z - y + x*z + 1
o7 : R</pre>
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This is a different answer from the SINGULAR book, since the choice of divisor affects the answer.<table class="examples"><tr><td><pre>i8 : f % (forceGB matrix{{f2,f1}})
2 3
o8 = y z + z
o8 : R</pre>
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