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<head><title>toricGraver -- calculates the Graver basis of the toric ideal; invokes "graver" from 4ti2</title>
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<div><h1>toricGraver -- calculates the Graver basis of the toric ideal; invokes "graver" 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>toricGraver(A) or toricGraver(A,R)</tt></div>
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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>
<li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, polynomial ring in which the toric ideal <i>I<sub>A</sub></i> should live</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>B</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose rows give binomials that form the Graver basis of the toric ideal of <tt>A</tt>, or</span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, whose generators form the Graver basis for the toric ideal</span></li>
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<div class="single"><h2>Description</h2>
<div><div>The Graver basis for any toric ideal <i>I<sub>A</sub></i> contains (properly) the union of all reduced Groebner basis of <i>I<sub>A</sub></i>. Any element in the Graver basis of the ideal is called a primitive binomial.</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 : toricGraver(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.
-------------------------------------------------
Final basis has 1 inhomogeneous, 10 homogeneous and 0 free elements.
Writing 5 vectors to graver file, with respect to symmetry.
4ti2 Total Time: 0.00 secs
using temporary file name /tmp/M2-3969-1
o2 = | 1 -2 1 0 |
| 2 -3 0 1 |
| 1 -1 -1 1 |
| 0 1 -2 1 |
| 1 0 -3 2 |
5 4
o2 : Matrix ZZ <--- ZZ</pre>
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<div>If we prefer to store the ideal instead, we may use:</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 : toricGraver(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.
-------------------------------------------------
Final basis has 1 inhomogeneous, 10 homogeneous and 0 free elements.
Writing 5 vectors to graver file, with respect to symmetry.
4ti2 Total Time: 0.00 secs
using temporary file name /tmp/M2-3969-2
2 3 2 2 3 2
o4 = ideal (- b + a*c, - b + a d, - b*c + a*d, - c + b*d, - c + a*d )
o4 : Ideal of R</pre>
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<div>Note that this last ideal equals the toric ideal <i>I<sub>A</sub></i> since every Graver basis element is actually in <i>I<sub>A</sub></i>.</div>
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<div class="waystouse"><h2>Ways to use <tt>toricGraver</tt> :</h2>
<ul><li>toricGraver(Matrix)</li>
<li>toricGraver(Matrix,Ring)</li>
</ul>
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