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<head><title>hilbertFunct -- return the Hilbert function of a polynomial ring mod a homogeneous ideal as a list</title>
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<div><h1>hilbertFunct -- return the Hilbert function of a polynomial ring mod a homogeneous ideal as a list</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>L=hilbertFunct I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a homogeneous ideal in a polynomial ring or a quotient of a polynomial ring (where the ring has the standard grading)</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, returns the Hilbert function of <tt>(ring I)/I</tt> as a list</span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_hilbert__Funct_lp..._cm_sp__Max__Degree_sp_eq_gt_sp..._rp.html">MaxDegree => ...</a>, -- bound degree through which Hilbert function is computed</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Let <tt>I</tt> be a homogeneous ideal in a ring <tt>R</tt> that is either a polynomial ring or a quotient of a polynomial ring, and suppose that <tt>R</tt> has the standard grading. <tt>hilbertFunct</tt> returns the Hilbert function of <tt>R/I</tt> as a list.</p>
<p>If <tt>R/I</tt> is Artinian, then the default is for <tt>hilbertFunct</tt> to return the entire Hilbert function (i.e., until the Hilbert function is zero) of <tt>R/I</tt> as a list. The user can override this by using the <tt>MaxDegree</tt> option to bound the highest degree considered.</p>
<p>If <tt>R/I</tt> is not Artinian, then <tt>hilbertFunct</tt> returns the Hilbert function of <tt>R/I</tt> through degree 20. Again, the user can select a different upper bound for the degree by using the <tt>MaxDegree</tt> option.</p>
<div>We require the standard grading on <tt>R</tt> in order to compute with the Hilbert series, which is presently much faster than repeatedly computing the Hilbert function.</div>
<table class="examples"><tr><td><pre>i1 : R=ZZ/32003[a..c];</pre>
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<tr><td><pre>i2 : hilbertFunct ideal(a^3,b^3,c^3)
o2 = {1, 3, 6, 7, 6, 3, 1}
o2 : List</pre>
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<tr><td><pre>i3 : hilbertFunct ideal(a^3,a*b^2)
o3 = {1, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
------------------------------------------------------------------------
24, 25}
o3 : List</pre>
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<tr><td><pre>i4 : hilbertFunct(ideal(a^3,a*b^2),MaxDegree=>4)
o4 = {1, 3, 6, 8, 9}
o4 : List</pre>
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<tr><td><pre>i5 : M=ideal(a^3,b^4,a*c);
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : Q=R/M;</pre>
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<tr><td><pre>i7 : hilbertFunct ideal(c^4)
o7 = {1, 3, 5, 6, 5, 3, 1}
o7 : List</pre>
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<tr><td><pre>i8 : hilbertFunct ideal(b*c,a*b)
o8 = {1, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
o8 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_is__H__F.html" title="is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal">isHF</a> -- is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal</span></li>
<li><span><a href="_lex__Ideal.html" title="produce a lexicographic ideal">lexIdeal</a> -- produce a lexicographic ideal</span></li>
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<div class="waystouse"><h2>Ways to use <tt>hilbertFunct</tt> :</h2>
<ul><li>hilbertFunct(Ideal)</li>
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