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<head><title>ehrhartRing -- Ehrhart ring</title>
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<div><h1>ehrhartRing -- Ehrhart ring</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>ehrhartRing(I)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the leading monomials of the elements of the ideal are considered as generators of a lattice polytope</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list containing one or two ideals</span></li>
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<div class="single"><h2>Description</h2>
<div>The exponent vectors of the leading monomials of the elements of <tt>I</tt> are considered as generators of a lattice polytope. The function returns a list of ideals:<br/><br/><em>(i)</em> If the last ring variable is not used by the monomials, it is treated as the auxiliary variable of the Ehrhart ring. The function returns two ideals, the first containing the monomials representing the lattice points of the polytope, the second containing the generators of the Ehrhart ring.<br/><br/><em>(ii)</em> If the last ring variable is used by the monomials, the function returns only one ideal, namely the monomials representing the lattice points of the polytope.<table class="examples"><tr><td><pre>i1 : R=ZZ/37[x,y,t];</pre>
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<tr><td><pre>i2 : I=ideal(x^3, x^2*y, y^3, x*y^2);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : ehrhartRing(I)
3 2 2 3 3 2 2 3
o3 = {ideal (x , x y, x*y , y ), ideal (x t, x y*t, x*y t, y t)}
o3 : List</pre>
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<tr><td><pre>i4 : J=I+ideal(x*y^2*t^7);
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : ehrhartRing(J)
2 7 2 6 2 5 2 4 2 3 2 3 2 2 2 2
o5 = {ideal (x*y t , x*y t , x*y t , x*y t , x y*t , x*y t , x y*t , x*y t ,
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2 2 3 2 2 3
x y*t, x*y t, x , x y, x*y , y )}
o5 : List</pre>
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<div class="waystouse"><h2>Ways to use <tt>ehrhartRing</tt> :</h2>
<ul><li>ehrhartRing(Ideal)</li>
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