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<head><title>hilbertPolynomial(ProjectiveVariety) -- compute the Hilbert polynomial of the projective variety</title>
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<div><h1>hilbertPolynomial(ProjectiveVariety) -- compute the Hilbert polynomial of the projective variety</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 V</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>V</tt>, <span>a <a href="___Projective__Variety.html">projective variety</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 an example of the <a href="_hilbert__Polynomial.html">Hilbert polynomial</a> of a projective Hilbert variety. This is the same as the Hilbert polynomial of its coordinate ring.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : I = monomialCurveIdeal(R, {1,3,4});
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : V = Proj(R/I)
o3 = V
o3 : ProjectiveVariety</pre>
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<tr><td><pre>i4 : h = hilbertPolynomial V
o4 = - 3*P + 4*P
0 1
o4 : ProjectiveHilbertPolynomial</pre>
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<tr><td><pre>i5 : hilbertPolynomial(V, Projective=>false)
o5 = 4i + 1
o5 : QQ[i]</pre>
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These Hilbert polynomials can serve as <a href="_hilbert__Function.html">Hilbert functions</a> too since the values of the Hilbert polynomial eventually are the same as the Hilbert function of the sheaf of rings or of the underlying ring.<table class="examples"><tr><td><pre>i6 : apply(5, k-> h(k))
o6 = {1, 5, 9, 13, 17}
o6 : List</pre>
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<tr><td><pre>i7 : apply(5, k-> hilbertFunction(k,ring V))
o7 = {1, 4, 9, 13, 17}
o7 : List</pre>
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