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<head><title>pureBetti(List) -- list of smallest integral Betti numbers corresponding to a degree sequence</title>
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<div><h1>pureBetti(List) -- list of smallest integral Betti numbers corresponding to a degree sequence</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>pureBetti L</tt></div>
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<li><span>Function: <a href="_pure__Betti_lp__List_rp.html" title="list of smallest integral Betti numbers corresponding to a degree sequence">pureBetti</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of strictly increasing integers</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of the minimal integral Betti numbers which satisfy the Herzog-Kuhl equations</span></li>
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<div class="single"><h2>Description</h2>
<div>The numerator P(t) of the Hilbert function of a module whose free resolution has a pure resolution of type L has the form P(t) = b_0 t^(d_0) - b_1 t^(d_1) + ... + (-1)^c b_c t^(d_c), where L = {d_0, ..., d_c}. If (1-t)^c divides P(t), as in the case where the module has codimension c, then the b_0, ..., b_c are determined up to a unique scalar multiple. This routine returns the smallest positive integral solution of these (Herzog-Kuhl) equations.<table class="examples"><tr><td><pre>i1 : pureBetti{0,2,4,5}
o1 = {3, 10, 15, 8}
o1 : List</pre>
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<tr><td><pre>i2 : pureBetti{0,3,4,5,6,7,10}
o2 = {1, 50, 175, 252, 175, 50, 1}
o2 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_pure__Betti__Diagram_lp__List_rp.html" title="pure Betti diagram given a list of degrees">pureBettiDiagram</a> -- pure Betti diagram given a list of degrees</span></li>
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