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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_sp1.8.4.html" title="elimination of variables">Singular Book 1.8.4</a></div>
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<div><h1>Singular Book 1.8.4 -- elimination of variables</h1>
<div>There are several methods to eliminate variables in Macaulay2.<table class="examples"><tr><td><pre>i1 : loadPackage "Elimination";</pre>
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<tr><td><pre>i2 : A = QQ[t,x,y,z];</pre>
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<tr><td><pre>i3 : I = ideal"t2+x2+y2+z2,t2+2x2-xy-z2,t+y3-z3";
o3 : Ideal of A</pre>
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<tr><td><pre>i4 : eliminate(I,t)
2 2 2 6 3 3 6 2 2
o4 = ideal (x - x*y - y - 2z , y - 2y z + z + x*y + 2y + 3z )
o4 : Ideal of A</pre>
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Alternatively, one may do it by hand: the elements of the Groebner basis under an elimination order not involving <tt>t</tt> generate the elimination ideal.<table class="examples"><tr><td><pre>i5 : A1 = QQ[t,x,y,z,MonomialOrder=>{1,3}];</pre>
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<tr><td><pre>i6 : I = substitute(I,A1);
o6 : Ideal of A1</pre>
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<tr><td><pre>i7 : transpose gens gb I
o7 = {-2} | x2-xy-y2-2z2 |
{-6} | y6-2y3z3+z6+xy+2y2+3z2 |
{-3} | t+y3-z3 |
3 1
o7 : Matrix A1 <--- A1</pre>
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Here is another elimination ideal. Weights not given are assumed to be zero.<table class="examples"><tr><td><pre>i8 : A2 = QQ[t,x,y,z,MonomialOrder=>Weights=>{1}];</pre>
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<tr><td><pre>i9 : I = substitute(I,A2);
o9 : Ideal of A2</pre>
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<tr><td><pre>i10 : transpose gens gb I
o10 = {-2} | x2-xy-y2-2z2 |
{-6} | y6-2y3z3+z6+xy+2y2+3z2 |
{-3} | t+y3-z3 |
3 1
o10 : Matrix A2 <--- A2</pre>
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The same order as the previous one:<table class="examples"><tr><td><pre>i11 : A3 = QQ[t,x,y,z,MonomialOrder=>Eliminate 1];</pre>
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<tr><td><pre>i12 : I = substitute(I,A3);
o12 : Ideal of A3</pre>
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<tr><td><pre>i13 : transpose gens gb I
o13 = {-2} | x2-xy-y2-2z2 |
{-6} | y6-2y3z3+z6+xy+2y2+3z2 |
{-3} | t+y3-z3 |
3 1
o13 : Matrix A3 <--- A3</pre>
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