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<head><title>radical -- the radical of an ideal</title>
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<div><h1>radical -- the radical of an ideal</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>radical I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, the radical of <tt>I</tt> .</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Unmixed => </tt><span><span>a <a href="___Boolean.html">Boolean value</a></span>, <span>default value false</span>, whether it is known that the ideal <tt>I</tt> is unmixed. The ideal <tt>I</tt> is said to be unmixed if all associated primes of <tt>R/I</tt> have the same dimension. In this case the algorithm tends to be much faster.</span></span></li>
<li><span><tt>Strategy => </tt><span><span>default value Decompose</span>, the strategy to use, either <tt>Decompose</tt> or <tt>Unmixed</tt></span></span></li>
<li><span><tt>CompleteIntersection => </tt><span><span>an <a href="___Ideal.html">ideal</a></span>, <span>default value null</span>, an ideal <tt>J</tt> of the same height as <tt>I</tt> whose generators form a maximal regular sequence contained in <tt>I</tt>. Providing this option as a hint allows a separate, often faster, algorithm to be used to compute the radical. This option should only be used if <tt>J</tt> is nice in some way. For example, if <tt>J</tt> is randomly generated, but <tt>I</tt> is relatively sparse, then this will most likely run slower than just giving the <a href="___Unmixed.html" title="name for an optional argument">Unmixed</a> option.</span></span></li>
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<div class="single"><h2>Description</h2>
<div>If I is an ideal in an affine ring (i.e. a quotient of a polynomial ring over a field), and if the characteristic of this field is large enough (see below), then this routine yields the radical of the ideal I.<p/>
The method used is the Eisenbud-Huneke-Vasconcelos algorithm. See their paper in Inventiones Mathematicae, 1993, for more details on the algorithm.<p/>
The algorithms used generally require that the characteristic of the ground field is larger than the degree of each primary component. In practice, this means that if the characteristic is something like 32003, rather than, for example, 5, the methods used will produce the radical of <tt>I</tt>. Of course, you may do the computation over <a href="___Q__Q.html" title="the class of all rational numbers">QQ</a>, but it will often run much slower. In general, this routine still needs to be tuned for speed.<p/>
Computes the radical of <tt>I</tt> using the Eisenbud-Huneke-Vasconcelos algorithm. If <tt>I</tt> is <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span>, a faster "combinatorial" algorithm is used.<table class="examples"><tr><td><pre>i1 : R=QQ[x,y]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I=ideal((x^2+1)^2*y, y+1)
4 2
o2 = ideal (x y + 2x y + y, y + 1)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : radical I
2
o3 = ideal (- y - 1, - x - 1)
o3 : Ideal of R</pre>
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For another example, see <a href="_component_spexample.html" title="">component example</a>.</div>
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<div class="single"><h2>Caveat</h2>
<div>The current implementation requires that the characteristic of the ground field is either zero or a "large" prime (unless <tt>I</tt> is <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span>).</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_minimal__Primes.html" title="minimal associated primes of an ideal">minimalPrimes</a> -- minimal associated primes of an ideal</span></li>
<li><span><a href="_top__Components.html" title="compute top dimensional component">topComponents</a> -- compute top dimensional component</span></li>
<li><span><a href="_remove__Lowest__Dimension.html" title="remove components of lowest dimension">removeLowestDimension</a> -- remove components of lowest dimension</span></li>
<li><span><a href="_saturate.html" title="saturation of ideal or submodule">saturate</a> -- saturation of ideal or submodule</span></li>
<li><span><a href="_quotient.html" title="quotient or division">quotient</a> -- quotient or division</span></li>
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<div class="waystouse"><h2>Ways to use <tt>radical</tt> :</h2>
<ul><li>radical(Ideal)</li>
<li>radical(MonomialIdeal)</li>
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