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<head><title>toricCircuits -- calculates the circuits of the toric ideal; invokes "circuits" from 4ti2</title>
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<div><h1>toricCircuits -- calculates the circuits of the toric ideal; invokes "circuits" from 4ti2</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>toricCircuits(A)</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose columns parametrize the toric variety. The toric ideal <i>I<sub>A</sub></i> is the kernel of the map defined by <tt>A</tt></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>B</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose rows form the circuits of A</span></li>
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<div class="single"><h2>Description</h2>
<div><div>The circuits are contained in the Graver basis of <i>I<sub>A</sub></i>. In fact, they are precisely the primitive binomials in the ideal with minimal support.</div>
<table class="examples"><tr><td><pre>i1 : A = matrix "1,1,1,1; 1,2,3,4"
o1 = | 1 1 1 1 |
| 1 2 3 4 |
2 4
o1 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i2 : C = toricCircuits A
-------------------------------------------------
4ti2 version 1.3.2, Copyright (C) 2006 4ti2 team.
4ti2 comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome
to redistribute it under certain conditions.
For details, see the file COPYING.
-------------------------------------------------
Using 64 bit integers.
4ti2 Total Time: 0.00 secs.
using temporary file name /tmp/M2-3752-1
o2 = | 0 1 -2 1 |
| 1 -2 1 0 |
| 1 0 -3 2 |
| 2 -3 0 1 |
4 4
o2 : Matrix ZZ <--- ZZ</pre>
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<div>The ideal generated by the circuits of A in general differs from the toric ideal of A. For example:</div>
<table class="examples"><tr><td><pre>i3 : R = QQ[a..d]
o3 = R
o3 : PolynomialRing</pre>
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<tr><td><pre>i4 : Icircuit = toBinomial(toricCircuits(A), R) -- this is the circuit ideal of A
-------------------------------------------------
4ti2 version 1.3.2, Copyright (C) 2006 4ti2 team.
4ti2 comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome
to redistribute it under certain conditions.
For details, see the file COPYING.
-------------------------------------------------
Using 64 bit integers.
4ti2 Total Time: 0.00 secs.
using temporary file name /tmp/M2-3752-2
2 2 3 2 3 2
o4 = ideal (- c + b*d, - b + a*c, - c + a*d , - b + a d)
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : I = toBinomial(toricMarkov(A), R)
-------------------------------------------------
4ti2 version 1.3.2, Copyright (C) 2006 4ti2 team.
4ti2 comes with ABSOLUTELY NO WARRANTY.
This is free software, and you are welcome
to redistribute it under certain conditions.
For details, see the file COPYING.
-------------------------------------------------
Using 64 bit integers.
4ti2 Total Time: 0.00 secs.
using temporary file name /tmp/M2-3752-3
2 2
o5 = ideal (- c + b*d, - b + a*c, - b*c + a*d)
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : I==Icircuit
o6 = false</pre>
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<div>The two ideals are not the same. There is a minimal generator of I which is not a circuit:</div>
<table class="examples"><tr><td><pre>i7 : a*d-b*c % I -- this binomial is in I:
o7 = 0
o7 : R</pre>
</td></tr>
<tr><td><pre>i8 : a*d-b*c % Icircuit -- but not in Icircuit:
o8 = - b*c + a*d
o8 : R</pre>
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<div class="waystouse"><h2>Ways to use <tt>toricCircuits</tt> :</h2>
<ul><li>toricCircuits(Matrix)</li>
</ul>
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