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<head><title>hilbertSeries(ProjectiveHilbertPolynomial) -- compute the Hilbert series of a projective Hilbert polynomial</title>
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<div><h1>hilbertSeries(ProjectiveHilbertPolynomial) -- compute the Hilbert series of a projective Hilbert 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>hilbertSeries P</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>P</tt>, <span>a <a href="___Projective__Hilbert__Polynomial.html">projective Hilbert polynomial</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 projective Hilbert polynomial.<table class="examples"><tr><td><pre>i1 : P = projectiveHilbertPolynomial 3
o1 = P
3
o1 : ProjectiveHilbertPolynomial</pre>
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<tr><td><pre>i2 : s = hilbertSeries P
1
o2 = --------
4
(1 - T)
o2 : Expression of class Divide</pre>
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<tr><td><pre>i3 : numerator s
o3 = 1
o3 : ZZ[T]</pre>
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Computing the <a href="_hilbert__Series.html">Hilbert series</a> of a projective variety can be useful for finding the h-vector of a simplicial complex from its f-vector. For example, consider the octahedron. The ideal below is its Stanley-Reisner ideal. We can see its f-vector (1, 6, 12, 8) in the Hilbert polynomial, and then we get the h-vector (1,3,3,1) from the coefficients of the Hilbert series projective Hilbert polynomial.<table class="examples"><tr><td><pre>i4 : R = QQ[a..h];</pre>
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<tr><td><pre>i5 : I = ideal (a*b, c*d, e*f);
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : P=hilbertPolynomial(I)
o6 = - P + 6*P - 12*P + 8*P
1 2 3 4
o6 : ProjectiveHilbertPolynomial</pre>
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<tr><td><pre>i7 : s = hilbertSeries P
2 3
1 + 3T + 3T + T
o7 = -----------------
5
(1 - T)
o7 : Expression of class Divide</pre>
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<tr><td><pre>i8 : numerator s
2 3
o8 = 1 + 3T + 3T + T
o8 : ZZ[T]</pre>
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