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<head><title>toricMarkov -- calculates a generating set of the toric ideal I_A, given A; invokes "markov" from 4ti2</title>
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<div><h1>toricMarkov -- calculates a generating set of the toric ideal I_A, given A; invokes "markov" 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>toricMarkov(A) or toricMarkov(A, InputType => "lattice") or toricMarkov(A,R)</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 is the kernel of the map defined by <tt>A</tt>. Otherwise, if InputType is set to "lattice", the rows of <tt>A</tt> are a lattice basis and the toric ideal is the saturation of the lattice basis ideal.</span></li>
<li><span><tt>InputType => </tt><span><tt>s</tt>, <span>default value null</span>, which is the string "lattice" if rows of <tt>A</tt> specify a lattice basis</span></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>
</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 a Markov Basis of the lattice <i>{z integral : A z = 0}</i> or the lattice spanned by the rows of <tt>A</tt> if the option <tt>InputType => "lattice"</tt> is used</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__Markov.html">InputType => ...</a>, -- calculates a generating set of the toric ideal I_A, given A; invokes "markov" from 4ti2</span></li>
</ul>
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<div class="single"><h2>Description</h2>
<div><div>Suppose we would like to comput the toric ideal defining the variety parametrized by the following matrix:</div>
<table class="examples"><tr><td><pre>i1 : A = matrix"1,1,1,1;0,1,2,3"
o1 = | 1 1 1 1 |
| 0 1 2 3 |
2 4
o1 : Matrix ZZ <--- ZZ</pre>
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<div>Since there are 4 columns, the ideal will live in the polynomial ring with 4 variables.</div>
<table class="examples"><tr><td><pre>i2 : R = QQ[a..d]
o2 = R
o2 : PolynomialRing</pre>
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<tr><td><pre>i3 : M = toricMarkov(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/M2-3657-1
o3 = | 0 1 -2 1 |
| 1 -2 1 0 |
| 1 -1 -1 1 |
3 4
o3 : Matrix ZZ <--- ZZ</pre>
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<div>Note that rows of M are the exponents of minimal generators of <i>I<sub>A</sub></i>. To get the ideal, we can do the following:</div>
<table class="examples"><tr><td><pre>i4 : I = toBinomial(M,R)
2 2
o4 = ideal (- c + b*d, - b + a*c, - b*c + a*d)
o4 : Ideal of R</pre>
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<div>Alternately, we might wish to give a lattice basis ideal instead of the matrix A. The lattice basis will be specified by a matrix, as follows:</div>
<table class="examples"><tr><td><pre>i5 : B = syz A
o5 = | -1 2 |
| 2 -3 |
| -1 0 |
| 0 1 |
4 2
o5 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i6 : N = toricMarkov(transpose B, InputType => "lattice")
-------------------------------------------------
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/M2-3657-2
o6 = | 0 1 -2 1 |
| 1 -2 1 0 |
| 1 -1 -1 1 |
3 4
o6 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i7 : J = toBinomial(N,R) -- toricMarkov(transpose B, R, InputType => "lattice")
2 2
o7 = ideal (- c + b*d, - b + a*c, - b*c + a*d)
o7 : Ideal of R</pre>
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<div>We can see that the two ideals are equal:</div>
<table class="examples"><tr><td><pre>i8 : I == J
o8 = true</pre>
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<div>Also, notice that instead of the sequence of commands above, we could have used the following:</div>
<table class="examples"><tr><td><pre>i9 : 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/M2-3657-3
2 2
o9 = ideal (- c + b*d, - b + a*c, - b*c + a*d)
o9 : Ideal of R</pre>
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<div class="waystouse"><h2>Ways to use <tt>toricMarkov</tt> :</h2>
<ul><li>toricMarkov(Matrix)</li>
<li>toricMarkov(Matrix,Ring)</li>
</ul>
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