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<head><title>multIdeal -- compute a multiplier ideal</title>
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<div><h1>multIdeal -- compute a multiplier 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>multIdeal(s,A) or multIdeal(s,A,m)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="___Arrangement.html">hyperplane arrangement</a></span>, a hyperplane arrangement</span></li>
<li><span><tt>s</tt>, <span>a <a href="../../Macaulay2Doc/html/___R__R.html">real number</a></span>, a real number</span></li>
<li><span><tt>m</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, optional list of positive integer multiplicities</span></li>
</ul>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the multiplier ideal of the arrangement at the value<tt>s</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The multiplier ideals of an given ideal depend on a nonnegative real parameter. This method computes the multiplier ideals of the defining ideal of a hyperplane arrangement, optionally with multiplicities <tt>m</tt>. This uses the explicit formula of M. Mustata [TAMS 358 (2006), no 11, 5015--5023], as simplified by Z. Teitler [PAMS 136 (2008), no 5, 1902--1913].<p/>
One can compute directly:<table class="examples"><tr><td><pre>i1 : A = typeA(3);</pre>
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<tr><td><pre>i2 : hilbertSeries multIdeal(3,A)
18
1 - T
o2 = --------
4
(1 - T)
o2 : Expression of class Divide</pre>
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Since the multiplier ideal is a locally constant function of its real parameter, one test to see at what values it changes:<table class="examples"><tr><td><pre>i3 : H = new MutableHashTable
o3 = MutableHashTable{}
o3 : MutableHashTable</pre>
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<tr><td><pre>i4 : scan(40,i -> (
s := i/20.;
I := multIdeal(s,A);
if not H#?I then H#I = {s} else H#I = H#I|{s}));</pre>
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<tr><td><pre>i5 : netList sort values H -- values of s giving same multiplier ideal
+---+----+---+----+---+----+---+----+---+----+---+----+---+----+
o5 = |0 |.05 |.1 |.15 |.2 |.25 |.3 | | | | | | | |
+---+----+---+----+---+----+---+----+---+----+---+----+---+----+
|.35|.4 |.45|.5 |.55|.6 |.65| | | | | | | |
+---+----+---+----+---+----+---+----+---+----+---+----+---+----+
|.7 |.75 |.8 |.85 |.9 |.95 | | | | | | | | |
+---+----+---+----+---+----+---+----+---+----+---+----+---+----+
|1 |1.05|1.1|1.15|1.2|1.25|1.3|1.35|1.4|1.45|1.5|1.55|1.6|1.65|
+---+----+---+----+---+----+---+----+---+----+---+----+---+----+
|1.7|1.75|1.8|1.85|1.9|1.95| | | | | | | | |
+---+----+---+----+---+----+---+----+---+----+---+----+---+----+</pre>
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<div class="waystouse"><h2>Ways to use <tt>multIdeal</tt> :</h2>
<ul><li>multIdeal(Number,Arrangement)</li>
<li>multIdeal(Number,Arrangement,List)</li>
<li>multIdeal(RR,Arrangement)</li>
<li>multIdeal(RR,Arrangement,List)</li>
</ul>
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