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<head><title>FourTiTwo -- Interface for 4ti2</title>
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<div><h1>FourTiTwo -- Interface for 4ti2</h1>
<div class="single"><h2>Description</h2>
<div><p>Interfaces most of the functionality of the software <tt>4ti2</tt> available at <a href="http://www.4ti2.de/">http://www.4ti2.de/</a>. (The user needs to have <tt>4ti2</tt> installed on his/her machine.)</p>
<p>A <i>d×n</i> integral matrix <i>A</i> (with nonnegative entries) specifies a map from a polynomial ring in d variables to a polynomial ring with n variables by specifying exponents of the variables indexing its columns. For example, if <i>A</i> is a matrix <p align=center><i>
<table class="matrix" border=1><tr><td><table><tr><td> 3</td><td>2</td><td>1</td><td>0</td></tr><tr><td> 0</td><td>1</td><td>2</td><td>3 </td></tr></table></td></tr></table>
</i></p> the map from <i>k[s,t]</i> to <i>k[a,b,c,d]</i> is given by <i>(s,t) → (s<sup>3</sup>,s<sup>2</sup>t,st<sup>2</sup>,t<sup>3</sup>)</i>.</p>
<p>The toric ideal <i>I<sub>A</sub></i> is the kernel of this map. It is minimally generated by the 2-minors of the matrix <p align=center><i>
<table class="matrix" border=1><tr><td><table><tr><td> x</td><td>y</td><td>z</td></tr><tr><td> y</td><td>z</td><td>w </td></tr></table></td></tr></table>
</i></p> Given the matrix <i>A</i>, one can compute its lattice basis ideal specified by the integral basis of the lattice <i>A</i>, the toric ideal <i>I<sub>A</sub></i>, its Groebner bases, etc. In practice, however, these are nontrivial computational tasks. The software <tt>4ti2</tt> is very efficient in computing these objects.</p>
<p>For more theoretical details (and more generality), see the standard reference: B. Sturmfels, <b>Gröbner bases and convex polytopes.</b> American Mathematical Society, University Lectures Series, No 8, Providence, Rhode Island, 1996.</p>
<p><b>Note for cygwin users:</b> If a problem occurs during package installation and/or loading, it should be fixed by setting the path inside the file <tt>.Macaulay2/init-FourTiTwo.m2</tt> to whatever folder <tt>4ti2</tt> is installed. For example, if <tt>4ti2</tt> has been installed in <tt>C:/cygwin/4ti2/win32</tt>, then the line inside the <tt>init-FourTiTwo.m2</tt> file will look like this: <tt>"path" => "C:/cygwin/4ti2/win32/"</tt> . Alternately, the path for <tt>4ti2</tt> may be set when loading the package using the following command: loadPackage("FourTiTwo", Configuration=>"path"=>"C:/cygwin/4ti2/win32/") assuming that 4ti2 has been installed in C:/cygwin/4ti2/win32.</p>
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<div class="single"><h2>Caveat</h2>
<div><div>If the package SimpleDoc is not found when installing <tt>FourTiTwo.m2</tt>, see questions and answers 6, 7, and 8 on the <i>Macaulay2</i> web site.</div>
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<div class="single"><h2>Authors</h2>
<ul><li><div class="single">Mike Stillman<span> <<a href="mailto:mike@math.cornell.edu">mike@math.cornell.edu</a>></span></div>
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<li><div class="single">Josephine Yu<span> <<a href="mailto:jyu@math.mit.edu">jyu@math.mit.edu</a>></span></div>
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<li><div class="single">Sonja Petrovic<span> <<a href="mailto:petrovic@math.uic.edu">petrovic@math.uic.edu</a>></span></div>
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<div class="single"><h2>Version</h2>
This documentation describes version <b>1.0</b> of FourTiTwo.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/FourTiTwo.m2">FourTiTwo.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Functions<ul><li><span><a href="_get__Matrix.html" title="reads a matrix from a 4ti2-formatted input file">getMatrix</a> -- reads a matrix from a 4ti2-formatted input file</span></li>
<li><span><a href="_hilbert__Basis.html" title="calculates the Hilbert basis of the cone; invokes "hilbert" from 4ti2">hilbertBasis</a> -- calculates the Hilbert basis of the cone; invokes "hilbert" from 4ti2</span></li>
<li><span><a href="_put__Matrix.html" title="writes a matrix into a file formatted for 4ti2">putMatrix</a> -- writes a matrix into a file formatted for 4ti2</span></li>
<li><span><a href="_rays.html" title="calculates the extreme rays of the cone; invokes "rays" from 4ti2">rays</a> -- calculates the extreme rays of the cone; invokes "rays" from 4ti2</span></li>
<li><span><a href="_to__Binomial.html" title="creates a toric ideal from a given set of exponents of its generators; equivalent to "4ti2-output --binomials" in 4ti2">toBinomial</a> -- creates a toric ideal from a given set of exponents of its generators; equivalent to "4ti2-output --binomials" in 4ti2</span></li>
<li><span><a href="_toric__Circuits.html" title="calculates the circuits of the toric ideal; invokes "circuits" from 4ti2">toricCircuits</a> -- calculates the circuits of the toric ideal; invokes "circuits" from 4ti2</span></li>
<li><span><a href="_toric__Graver.html" title="calculates the Graver basis of the toric ideal; invokes "graver" from 4ti2">toricGraver</a> -- calculates the Graver basis of the toric ideal; invokes "graver" from 4ti2</span></li>
<li><span><a href="_toric__Graver__Degrees.html" title="displays the degrees of all Graver basis elements for the toric ideal I_A; equivalent to "4ti2-output --degrees foo.gra" in 4ti2">toricGraverDegrees</a> -- displays the degrees of all Graver basis elements for the toric ideal I_A; equivalent to "4ti2-output --degrees foo.gra" in 4ti2</span></li>
<li><span><a href="_toric__Groebner.html" title="calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2">toricGroebner</a> -- calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2</span></li>
<li><span><a href="_toric__Markov.html" title="calculates a generating set of the toric ideal I_A, given A; invokes "markov" from 4ti2">toricMarkov</a> -- calculates a generating set of the toric ideal I_A, given A; invokes "markov" from 4ti2</span></li>
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<li><div class="single">Symbols<ul><li><span><a href="___Input__Type.html" title="">InputType</a></span></li>
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