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<head><title>statePolytope -- computes the state polytope of a homogeneous ideal</title>
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<div><h1>statePolytope -- computes the state polytope of a homogeneous ideal</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> P = statePolytope I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which must be homogeneous</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div><p/>
A <tt>statePolytope</tt> of an Ideal <tt>I</tt> has as normalFan
the Groebner fan of the ideal. We use the construction by Sturmfels, see Algorithm 3.2 in <a href="http://math.berkeley.edu/~bernd/index.html">Bernd Sturmfels'</a> <em>Groebner Bases and
Convex Polytopes</em>, volume 8 of University Lecture Series. American Mathematical Society,
first edition, 1995.<p/>
Consider the following ideal in a ring with 3 variables:<table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I = ideal (a-b,a-c,b-c)
o2 = ideal (a - b, a - c, b - c)
o2 : Ideal of R</pre>
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The state polytope of this ideal is a triangle in 3 space, because the ideal has three
initial ideals:<table class="examples"><tr><td><pre>i3 : statePolytope I
o3 = ({| b a |, | c b |, | c a |}, {ambient dimension => 3 })
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o3 : Sequence</pre>
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The generators of the three initial ideals are given in the first part of the result.</div>
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<div class="waystouse"><h2>Ways to use <tt>statePolytope</tt> :</h2>
<ul><li>statePolytope(Ideal)</li>
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