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<head><title>hilbertPolynomial(Ring) -- compute the Hilbert polynomial of the ring</title>
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<div><h1>hilbertPolynomial(Ring) -- compute the Hilbert polynomial of the ring</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 R</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>R</tt>, <span>a <a href="___Ring.html">ring</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 coordinate ring of the rational quartic curve in <tt>P^3.</tt><table class="examples"><tr><td><pre>i1 : R = ZZ/101[a..d];</pre>
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<tr><td><pre>i2 : S = coimage map(R, R, {a^4, a^3*b, a*b^3, b^4});</pre>
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<tr><td><pre>i3 : presentation S
o3 = | bc-ad c3-bd2 ac2-b2d b3-a2c |
1 4
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : h = hilbertPolynomial S
o4 = - 3*P + 4*P
0 1
o4 : ProjectiveHilbertPolynomial</pre>
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<tr><td><pre>i5 : hilbertPolynomial(S, Projective=>false)
o5 = 4i + 1
o5 : QQ[i]</pre>
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The rational quartic curve in <tt>P^3</tt> is therefore 'like' 4 copies of <tt>P^1</tt>, with three points missing. One can see this by noticing that there is a deformation of the rational quartic to the union of 4 lines, or 'sticks', which intersect in three successive points.<p/>
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. <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,S))
o7 = {1, 4, 9, 13, 17}
o7 : List</pre>
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