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<head><title>hilbertSeries(Module) -- compute the Hilbert series of the module</title>
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<div><h1>hilbertSeries(Module) -- compute the Hilbert series of the module</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>hilbertSeries M</tt></div>
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<li><span>Function: <a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</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="___Divide.html">divide expression</a></span>, the Hilbert series</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_hilbert__Series_lp..._cm_sp__Order_sp_eq_gt_sp..._rp.html">Order => ...</a>, -- display the truncated power series expansion</span></li>
<li><span><a href="_hilbert__Series_lp..._cm_sp__Reduce_sp_eq_gt_sp..._rp.html">Reduce => ...</a>, -- reduce the Hilbert series</span></li>
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<div class="single"><h2>Description</h2>
<div>We compute the <a href="_hilbert__Series.html">Hilbert series</a> of a module.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x, Degrees => {2}];</pre>
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<tr><td><pre>i2 : M = module ideal x^2
o2 = image | x2 |
1
o2 : R-module, submodule of R</pre>
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<tr><td><pre>i3 : s = hilbertSeries M
4
T
o3 = --------
2
(1 - T )
o3 : Expression of class Divide</pre>
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<tr><td><pre>i4 : numerator s
4
o4 = T
o4 : ZZ[T]</pre>
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<tr><td><pre>i5 : poincare M
4
o5 = T
o5 : ZZ[T]</pre>
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Recall that the variables of the power series are the variables of the <a href="_degrees__Ring.html">degrees ring</a>.<table class="examples"><tr><td><pre>i6 : R=ZZ/101[x, Degrees => {{1,1}}];</pre>
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<tr><td><pre>i7 : M = module ideal x^2;</pre>
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<tr><td><pre>i8 : s = hilbertSeries M
2 2
T T
0 1
o8 = ----------
(1 - T T )
0 1
o8 : Expression of class Divide</pre>
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<tr><td><pre>i9 : numerator s
2 2
o9 = T T
0 1
o9 : ZZ[T , T ]
0 1</pre>
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<tr><td><pre>i10 : poincare M
2 2
o10 = T T
0 1
o10 : ZZ[T , T ]
0 1</pre>
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