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<head><title>hilbertPolynomial(CoherentSheaf) -- compute the Hilbert polynomial of the coherent sheaf</title>
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<div><h1>hilbertPolynomial(CoherentSheaf) -- compute the Hilbert polynomial of the coherent sheaf</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>hilbertPolynomial S</tt></div>
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<li><span>Function: <a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>S</tt>, <span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Projective__Hilbert__Polynomial.html">projective Hilbert polynomial</a></span>, unless the option Projective is false</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__Polynomial_lp..._cm_sp__Projective_sp_eq_gt_sp..._rp.html">Projective => ...</a>, -- choose how to display the Hilbert polynomial</span></li>
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<div class="single"><h2>Description</h2>
<div>We compute the <a href="_hilbert__Polynomial.html">Hilbert polynomial</a> of a coherent sheaf.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x_0..x_2];</pre>
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<tr><td><pre>i2 : V = Proj R;</pre>
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<tr><td><pre>i3 : S = sheaf(image matrix {{x_0^3+x_1^3+x_2^3}})
o3 = image | x_0^3+x_1^3+x_2^3 |
1
o3 : coherent sheaf on V, subsheaf of OO
V</pre>
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<tr><td><pre>i4 : h = hilbertPolynomial S
o4 = 3*P - 3*P + P
0 1 2
o4 : ProjectiveHilbertPolynomial</pre>
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<tr><td><pre>i5 : hilbertPolynomial(S, Projective=>false)
1 2 3
o5 = -i - -i + 1
2 2
o5 : QQ[i]</pre>
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