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<head><title>independentSets -- some size-maximal independent subsets of variables modulo an ideal</title>
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<div><h1>independentSets -- some size-maximal independent subsets of variables modulo 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>independentSets J</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>J</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, or <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___List.html">list</a></span>, of products of variables. The support of any one of these products is a maximal independent subset of variables modulo <tt>J</tt></span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Limit => </tt><span><span>an <a href="___Z__Z.html">integer</a></span>, <span>default value infinity</span>, the maximum number of independent sets to be found</span></span></li>
</ul>
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</li>
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<div class="single"><h2>Description</h2>
<div>An independent set of variables of an ideal <tt>J</tt> in a polynomial ring <tt>R</tt> is a set of variables that are algebraically independent modulo <tt>J</tt> (i.e. there is no polynomial in <tt>J</tt> involving only these sets of variables.<p/>
If the Krull dimension of <tt>R/J</tt> is <tt>d</tt>, then a maximal independent set is an independent set having size <tt>d</tt>.<p/>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..h];</pre>
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<tr><td><pre>i2 : I = minors(2,genericMatrix(R,a,2,4))
o2 = ideal (- b*c + a*d, - b*e + a*f, - d*e + c*f, - b*g + a*h, - d*g + c*h,
------------------------------------------------------------------------
- f*g + e*h)
o2 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i3 : inI = ideal leadTerm I
o3 = ideal (f*g, d*g, b*g, d*e, b*e, b*c)
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : independentSets I
o4 = {a*b*d*f*h, a*c*d*f*h, a*c*e*f*h, a*c*e*g*h}
o4 : List</pre>
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<tr><td><pre>i5 : independentSets inI
o5 = {a*b*d*f*h, a*c*d*f*h, a*c*e*f*h, a*c*e*g*h}
o5 : List</pre>
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<p/>
The independent sets returned correspond one for one with the minimal primes of smallest codimension of the ideal of lead terms of J.<table class="examples"><tr><td><pre>i6 : I = ideal"abc,bcd,cde,adf,cgh,b3f,a3g"
3 3
o6 = ideal (a*b*c, b*c*d, c*d*e, a*d*f, c*g*h, b f, a g)
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : minimalPrimes I
o7 = {ideal (c, g, f), ideal (a, d, f, h), ideal (a, b, h, e), ideal (a, b,
------------------------------------------------------------------------
g, e), ideal (a, b, c), ideal (a, b, d, h), ideal (a, c, f), ideal (a,
------------------------------------------------------------------------
d, g, f), ideal (b, g, f, e), ideal (b, d, g)}
o7 : List</pre>
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<tr><td><pre>i8 : independentSets I
o8 = {a*b*d*e*h, a*c*e*f*h, b*d*e*g*h, d*e*f*g*h}
o8 : List</pre>
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The optional Limit argument is useful if you need only one, or several such independent sets.<table class="examples"><tr><td><pre>i9 : L = independentSets(I, Limit=>1)
o9 = {a*b*d*e*h}
o9 : List</pre>
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<p/>
Often, you want the list of the variables in a maximal independent set, or the list of those not in the set.<table class="examples"><tr><td><pre>i10 : support L_0
o10 = {a, b, d, e, h}
o10 : List</pre>
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<tr><td><pre>i11 : rsort toList(set gens R - set support L_0)
o11 = {c, f, g}
o11 : List</pre>
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<p/>
This function is useful as a subroutine to primary decomposition algorithms.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="../../PrimaryDecomposition/html/index.html" title="">PrimaryDecomposition</a></span></li>
<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="_support.html" title="list of variables occurring in a polynomial or matrix">support</a> -- list of variables occurring in a polynomial or matrix</span></li>
<li><span><a href="_rsort.html" title="sort a list or matrix in reverse order">rsort</a> -- sort a list or matrix in reverse order</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>independentSets</tt> :</h2>
<ul><li>independentSets(Ideal)</li>
<li>independentSets(MonomialIdeal)</li>
</ul>
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