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<head><title>toricGraverDegrees -- displays the degrees of all Graver basis elements for the toric ideal I_A; equivalent to "4ti2-output --degrees foo.gra" in 4ti2</title>
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<div><h1>toricGraverDegrees -- displays the degrees of all Graver basis elements for the toric ideal I_A; equivalent to "4ti2-output --degrees foo.gra" in 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>toricGraverDegrees(A)</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>
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<div class="single"><h2>Description</h2>
<div><div>Very often the Graver basis consits of too many binomials, and one is only interested in their degrees. In this case, instead of looking at the Graver basis of <i>I<sub>A</sub></i>, we may just want to look for the degrees of binomials which show up:</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>
</td></tr>
<tr><td><pre>i2 : toricGraver(A) -- the Graver basis
-------------------------------------------------
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/M2-3729-1
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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<tr><td><pre>i3 : toricGraverDegrees(A) -- just the degrees
-------------------------------------------------
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
-------------------------------------------------
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.
-------------------------------------------------
Norm = 4, number of elements = 3
Norm = 6, number of elements = 2
using temporary file name /tmp/M2-3729-2</pre>
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<p>Note that these are all 1-norms of the vectors. Since <i>I<sub>A</sub></i> is homogeneous, there are 3 binomials of degree 2 (norm 4) and 2 binomials of degree 3 (norm 6).</p>
<div>Here is a more complicated example, where one may not want to see the Graver basis elements explicitly. The toric ideal I<sub>M</sub> is the ideal of the rational normal scroll S(3,2,3):</div>
<table class="examples"><tr><td><pre>i4 : M = matrix "1,1,1,1,1,1,1,1,1,1,1; 1,1,1,1,0,0,0,0,0,0,0; 0,0,0,0,1,1,1,0,0,0,0; 0,0,0,0,0,0,0,1,1,1,1; 1,2,3,4,1,2,3,1,2,3,4"
o4 = | 1 1 1 1 1 1 1 1 1 1 1 |
| 1 1 1 1 0 0 0 0 0 0 0 |
| 0 0 0 0 1 1 1 0 0 0 0 |
| 0 0 0 0 0 0 0 1 1 1 1 |
| 1 2 3 4 1 2 3 1 2 3 4 |
5 11
o4 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i5 : toricGraverDegrees(M)
-------------------------------------------------
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, 486 homogeneous and 0 free elements.
Writing 243 vectors to graver file, with respect to symmetry.
4ti2 Total Time: 0.00 secs
-------------------------------------------------
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.
-------------------------------------------------
Norm = 4, number of elements = 37
Norm = 6, number of elements = 115
Norm = 8, number of elements = 76
Norm = 10, number of elements = 15
using temporary file name /tmp/M2-3729-3</pre>
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<div>Here is another example where with many Graver basis elements. The following matrix is a design matrix for a particular statistical model for a 4-node p1 network; see Fienberg-Petrovic-Rinaldo.</div>
<table class="examples"><tr><td><pre>i6 : A = matrix "1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0;0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0;0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,0,1;0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,1,1;0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0;1,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0;0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,1;0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,1,0,1;0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1";
9 18
o6 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i7 : toricGraverDegrees(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, 14924 homogeneous and 0 free elements.
Writing 7462 vectors to graver file, with respect to symmetry.
4ti2 Total Time: 1.36 secs
-------------------------------------------------
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.
-------------------------------------------------
Norm = 4, number of elements = 9
Norm = 6, number of elements = 67
Norm = 8, number of elements = 378
Norm = 10, number of elements = 792
Norm = 12, number of elements = 1800
Norm = 14, number of elements = 1872
Norm = 16, number of elements = 1224
Norm = 18, number of elements = 792
Norm = 20, number of elements = 432
Norm = 22, number of elements = 96
using temporary file name /tmp/M2-3729-4</pre>
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<div class="waystouse"><h2>Ways to use <tt>toricGraverDegrees</tt> :</h2>
<ul><li>toricGraverDegrees(Matrix)</li>
</ul>
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