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<head><title>poincare(Module) -- assemble degrees of an module into a polynomial</title>
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<div><h1>poincare(Module) -- assemble degrees of an module into a polynomial</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>poincare M</tt></div>
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<li><span>Function: <a href="_poincare.html" title="assemble degrees into polynomial">poincare</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Ring__Element.html">ring element</a></span>, in the Laurent polynomial ring whose variables correspond to the degrees of the ambient ring</span></li>
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<div class="single"><h2>Description</h2>
<div>We compute the <a href="_poincare.html">Poincare polynomial</a> of a module.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[w..z];</pre>
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<tr><td><pre>i2 : M = module monomialCurveIdeal(R,{1,3,4});</pre>
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<tr><td><pre>i3 : poincare M
2 3 4 5
o3 = T + 3T - 4T + T
o3 : ZZ[T]</pre>
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<tr><td><pre>i4 : numerator reduceHilbert hilbertSeries M
2 3 4 5
o4 = T + 3T - 4T + T
o4 : ZZ[T]</pre>
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Recall that the variables of the polynomial are the variables of the degrees ring.<table class="examples"><tr><td><pre>i5 : R=ZZ/101[x, Degrees => {{1,1}}];</pre>
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<tr><td><pre>i6 : M = module ideal x^2;</pre>
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<tr><td><pre>i7 : poincare M
2 2
o7 = T T
0 1
o7 : ZZ[T , T ]
0 1</pre>
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<tr><td><pre>i8 : numerator reduceHilbert hilbertSeries M
2 2
o8 = T T
0 1
o8 : ZZ[T , T ]
0 1</pre>
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