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<head><title>multUpperBound -- determine whether an ideal satisfies the upper bound of the multiplicity conjecture</title>
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<div><h1>multUpperBound -- determine whether an ideal satisfies the upper bound of the multiplicity conjecture</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>B=multUpperBound 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</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>B</tt>, <span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <tt>true</tt> if <tt>I</tt> satisfies the upper bound and <tt>false</tt> otherwise</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Let <tt>I</tt> be a homogeneous ideal of codimension <tt>c</tt> in a polynomial ring <tt>R</tt>. Huneke and Srinivasan (and later Herzog and Srinivasan in the non-Cohen-Macaulay case) conjectured that</p>
<p><tt>e(R/I) <= M<sub>1</sub> ... M<sub>c</sub> / c!</tt>,</p>
<p>where <tt>M<sub>i</sub></tt> is the maximum shift in the minimal graded free resolution of <tt>R/I</tt> at step <tt>i</tt>, and <tt>e(R/I)</tt> is the multiplicity of <tt>R/I</tt>. <tt>multUpperBound</tt> tests this inequality for the given ideal, returning <tt>true</tt> if the inequality holds and <tt>false</tt> otherwise, and it prints the upper bound and the multiplicity (listed as the degree).</p>
<div>This conjecture was proven in 2008 work of Eisenbud-Schreyer and Boij-Soderberg.</div>
<table class="examples"><tr><td><pre>i1 : R=ZZ/32003[a..c];</pre>
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<tr><td><pre>i2 : multUpperBound ideal(a^4,b^4,c^4)
degree = 64 upper bound = 64
o2 = true</pre>
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<tr><td><pre>i3 : multUpperBound ideal(a^3,b^5,c^6,a^2*b,a*b*c)
degree = 46 upper bound = 132
o3 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_cancel__All.html" title="make all potentially possible cancellations in the graded free resolution of an ideal">cancelAll</a> -- make all potentially possible cancellations in the graded free resolution of an ideal</span></li>
<li><span><a href="_mult__Upper__H__F.html" title="test a sufficient condition for the upper bound of the multiplicity conjecture">multUpperHF</a> -- test a sufficient condition for the upper bound of the multiplicity conjecture</span></li>
<li><span><a href="_mult__Lower__Bound.html" title="determine whether an ideal satisfies the lower bound of the multiplicity conjecture">multLowerBound</a> -- determine whether an ideal satisfies the lower bound of the multiplicity conjecture</span></li>
<li><span><a href="_mult__Bounds.html" title="determine whether an ideal satisfies the upper and lower bounds of the multiplicity conjecture">multBounds</a> -- determine whether an ideal satisfies the upper and lower bounds of the multiplicity conjecture</span></li>
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<div class="waystouse"><h2>Ways to use <tt>multUpperBound</tt> :</h2>
<ul><li>multUpperBound(Ideal)</li>
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