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<head><title>ehrhart -- calculates the Ehrhart polynomial of a polytope</title>
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<div><h1>ehrhart -- calculates the Ehrhart polynomial of a polytope</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>f = ehrhart P</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span>, which must be compact</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, Ehrhart polynomial as element of QQ[x]</span></li>
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<div class="single"><h2>Description</h2>
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<tt>ehrhart</tt> can only be applied to polytopes, i.e. compact polyhedra.
To calculate the Ehrhart polynomial, the number of lattice points in the first n
dilations of the polytope are calculated, where n is the dimension of the polytope.
A system of linear equations is then solved to find the polynomial.<table class="examples"><tr><td><pre>i1 : P = convexHull transpose matrix {{0,0,0},{1,0,0},{0,1,0},{1,1,3}}
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 4
number of rays => 0
number of vertices => 4
o1 : Polyhedron</pre>
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<tr><td><pre>i2 : ehrhart P
1 3 2 3
o2 = -x + x + -x + 1
2 2
o2 : QQ[x]</pre>
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<div class="waystouse"><h2>Ways to use <tt>ehrhart</tt> :</h2>
<ul><li>ehrhart(Polyhedron)</li>
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