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<head><title>simplex -- Simplex in the variables of a polynomial ring.</title>
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<div><h1>simplex -- Simplex in the variables of a polynomial 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>simplex(R)</tt><br/><tt>simplex(R,Rdual)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span></span></li>
<li><span><tt>Rdual</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Complex.html">embedded complex</a></span></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_compute__Faces.html">computeFaces => ...</a>, -- Compute faces of a simplex.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Returns a simplex on the variables of R.</p>
<p>If R does not have a coker grading then the standard projective space fan rays are added, see <a href="_add__Coker__Grading.html" title="Stores a cokernel grading in a polynomial ring.">addCokerGrading</a> and <a href="_rays__P__Pn.html" title="The rays of the standard fan of projective space.">raysPPn</a>.</p>
<p>The Option computeFaces=>false suppresses the computaton of all faces.</p>
<p>If Rdual is specified it is used for the vertices of the dual simplex, if not a new polynomial ring is created. It is graded by the coordinates of the vertices of the dual simplex.</p>
<p>The dual simplex is always created without face data.</p>
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<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : C=simplex(R)
o2 = 4: x x x x x
0 1 2 3 4
o2 : complex of dim 4 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0</pre>
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<tr><td><pre>i3 : grading C
o3 = | -1 -1 -1 -1 |
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
5 4
o3 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i4 : dC=C.dualComplex
o4 = 4: v v v v v
0 1 2 3 4
o4 : complex of dim 4 embedded in dim 4 (printing facets)
equidimensional, simplicial</pre>
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<tr><td><pre>i5 : grading dC
o5 = | -1 -1 -1 4 |
| -1 -1 4 -1 |
| -1 4 -1 -1 |
| 4 -1 -1 -1 |
| -1 -1 -1 -1 |
5 4
o5 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i6 : fc(dC);</pre>
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<tr><td><pre>i7 : dC
o7 = 4: v v v v v
0 1 2 3 4
o7 : complex of dim 4 embedded in dim 4 (printing facets)
equidimensional, simplicial, F-vector {1, 5, 10, 10, 5, 1}, Euler = 0</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_grading.html" title="The grading of a first order deformation or complex or polynomial ring.">grading</a> -- The grading of a first order deformation or complex or polynomial ring.</span></li>
<li><span><a href="_dual__Grading.html" title="The dual vertices of a polytope.">dualGrading</a> -- The dual vertices of a polytope.</span></li>
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<div class="waystouse"><h2>Ways to use <tt>simplex</tt> :</h2>
<ul><li>simplex(PolynomialRing)</li>
<li>simplex(PolynomialRing,PolynomialRing)</li>
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