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<head><title>circuits -- list the circuits of an arrangement</title>
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<div><h1>circuits -- list the circuits of an arrangement</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>circuits(A)</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>, hyperplane arrangement</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, A list of circuits of <tt>A</tt> each one expressed as a list of indices.</span></li>
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<div class="single"><h2>Description</h2>
<div>By definition, a circuit is a minimal set of hyperplanes with linearly dependent normal vectors.<table class="examples"><tr><td><pre>i1 : R := QQ[x,y,z];</pre>
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<tr><td><pre>i2 : A := arrangement {x,y,z,x-y,x-z,y-z};</pre>
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<tr><td><pre>i3 : L := circuits A
o3 = {{3, 4, 5}, {1, 2, 5}, {0, 2, 4}, {0, 1, 3}, {0, 1, 4, 5}, {0, 2, 3, 5},
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{1, 2, 3, 4}}
o3 : List</pre>
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<tr><td><pre>i4 : (C -> (tolist A)_C)\L
o4 = {{x - y, x - z, y - z}, {y, z, y - z}, {x, z, x - z}, {x, y, x - y}, {x,
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y, x - z, y - z}, {x, z, x - y, y - z}, {y, z, x - y, x - z}}
o4 : List</pre>
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An arrangement has circuits of length 2 if and only if it has repeated hyperplanes:<table class="examples"><tr><td><pre>i5 : A' := restriction(A,x)
o5 = {y, z, -y, -z, y - z}
o5 : Hyperplane Arrangement </pre>
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<tr><td><pre>i6 : circuits A'
o6 = {{1, 3}, {0, 2}, {2, 3, 4}, {0, 3, 4}, {1, 2, 4}, {0, 1, 4}}
o6 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_flats.html" title="list the flats of an arrangement of given rank">flats</a> -- list the flats of an arrangement of given rank</span></li>
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<div class="waystouse"><h2>Ways to use <tt>circuits</tt> :</h2>
<ul><li>circuits(Arrangement)</li>
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