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<div><h1>eliminate</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>eliminate(v,J)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>v</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span> or <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a variable or list of variables of a polynomial ring <tt>R</tt></span></li>
<li><span><tt>J</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in the ring <tt>R</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, generated by the elements of J not involving the variables v</span></li>
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<div class="single"><h2>Description</h2>
<div>If the ideal <tt>J</tt> is homogeneous, then an effort is made to use the Hilbert function to speed up the computation.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,a,b,c,d]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : f = x^2+a*x+b
2
o2 = x + x*a + b
o2 : R</pre>
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<tr><td><pre>i3 : g = x^2+c*x+d
2
o3 = x + x*c + d
o3 : R</pre>
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<tr><td><pre>i4 : time eliminate(x,ideal(f,g))
-- used 0.002999 seconds
2 2 2 2
o4 = ideal(a*b*c - b*c - a d + a*c*d - b + 2b*d - d )
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : time ideal resultant(f,g,x)
-- used 0.001 seconds
2 2 2 2
o5 = ideal(- a*b*c + b*c + a d - a*c*d + b - 2b*d + d )
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : sylvesterMatrix(f,g,x)
o6 = {0} | b 0 d 0 |
{1} | a b c d |
{2} | 1 a 1 c |
{3} | 0 1 0 1 |
4 4
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : discriminant(f,x)
2
o7 = - a + 4b
o7 : R</pre>
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<p>One may also switch the order of arguments: the ideal being the first argument. This usage has been deprecated, and should no longer be used.</p>
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<div class="single"><h2>Caveat</h2>
<div>The ring <tt>R</tt> should not be a quotient ring, or a non-commutative ring. Additionally, it would be nice to be able to use a DegreeLimit, or to be able to interrupt the computation.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html" title="">resultant</a></span></li>
<li><span><a href="_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html" title="">discriminant</a></span></li>
<li><span><a href="index.html" title="eliminating specified variables, and Sylvester resultant">Elimination</a> -- eliminating specified variables, and Sylvester resultant</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>eliminate</tt> :</h2>
<ul><li><span>eliminate(List,Ideal), see <span><a href="_eliminate.html" title="">eliminate</a></span></span></li>
<li><span>eliminate(RingElement,Ideal), see <span><a href="_eliminate.html" title="">eliminate</a></span></span></li>
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