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<head><title>hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal</title>
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<div><h1>hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal</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 I</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>I</tt>, <span>an <a href="___Ideal.html">ideal</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 the quotient of the ambient ring by an ideal.<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 : h = hilbertPolynomial I
o3 = - 3*P + 4*P
0 1
o3 : ProjectiveHilbertPolynomial</pre>
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<tr><td><pre>i4 : hilbertPolynomial (R/I)
o4 = - 3*P + 4*P
0 1
o4 : ProjectiveHilbertPolynomial</pre>
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<tr><td><pre>i5 : hilbertPolynomial(I, 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.<table class="examples"><tr><td><pre>i6 : apply(10, k-> h(k))
o6 = {1, 5, 9, 13, 17, 21, 25, 29, 33, 37}
o6 : List</pre>
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<tr><td><pre>i7 : apply(10, k-> hilbertFunction(k,I))
o7 = {1, 4, 9, 13, 17, 21, 25, 29, 33, 37}
o7 : List</pre>
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<div class="single"><h2>Caveat</h2>
<div>As is often the case, calling this function on an ideal <tt>I</tt> actually computes it for <tt>R/I</tt> where <tt>R</tt> is the ring of <tt>I</tt>.</div>
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