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<head><title>toricGroebner -- calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2</title>
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<div><h1>toricGroebner -- calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" 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>toricGroebner(A) or toricGroebner(A,R)</tt></div>
</dd></dl>
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</li>
<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>, ring with as least as many generators as the columns of <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 give binomials that form a Groebner basis of the toric ideal of <tt>A</tt></span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, whose generators form a Groebner basis for the toric ideal</span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_toric__Groebner.html">Weights => ...</a>, -- calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2</span></li>
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<div class="single"><h2>Description</h2>
<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 : toricGroebner(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/M2-3815-1
o2 = | -1 1 1 -1 |
| -1 2 -1 0 |
| 0 -1 2 -1 |
3 4
o2 : Matrix ZZ <--- ZZ</pre>
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<div>Note that the output of the command is a matrix whose rows are the exponents of the binomials that for a Groebner basis of the toric ideal <i>I<sub>A</sub></i>. As a shortcut, one can ask for the output to be an ideal instead:</div>
<table class="examples"><tr><td><pre>i3 : R = QQ[a..d]
o3 = R
o3 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i4 : toricGroebner(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/M2-3815-2
2 2
o4 = ideal (b*c - a*d, b - a*c, c - b*d)
o4 : Ideal of R</pre>
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<div><tt>4ti2</tt> offers the use of weight vectors representing term orders, as follows:</div>
<table class="examples"><tr><td><pre>i5 : toricGroebner(A,Weights=>{1,2,3,4})
-------------------------------------------------
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/M2-3815-3
o5 = | -1 1 1 -1 |
| -1 2 -1 0 |
| 0 -1 2 -1 |
3 4
o5 : Matrix ZZ <--- ZZ</pre>
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<div class="single"><h2>Caveat</h2>
<div><p>It seems that some versions of 4ti2 do not pick up on the weight vector. It may be better to run gb computation in M2 directly with specified weights.</p>
<p></p>
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<div class="waystouse"><h2>Ways to use <tt>toricGroebner</tt> :</h2>
<ul><li>toricGroebner(Matrix)</li>
<li>toricGroebner(Matrix,Ring)</li>
</ul>
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