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<head><title>multBounds -- determine whether an ideal satisfies the upper and lower bounds of the multiplicity conjecture</title>
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<div><h1>multBounds -- determine whether an ideal satisfies the upper and lower bounds 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=multBounds 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 both the upper and lower bounds hold 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> such that <tt>R/I</tt> is Cohen-Macaulay. Herzog, Huneke, and Srinivasan conjectured that if <tt>R/I</tt> is Cohen-Macaulay, then</p>
<p><tt>m<sub>1</sub> ... m<sub>c</sub> / c! <= e(R/I) <= M<sub>1</sub> ... M<sub>c</sub> / c!</tt>,</p>
<p>where <tt>m<sub>i</sub></tt> is the minimum shift in the minimal graded free resolution of <tt>R/I</tt> at step <tt>i</tt>, <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>. If <tt>R/I</tt> is not Cohen-Macaulay, the upper bound is still conjectured to hold. <tt>multBounds</tt> tests the inequalities for the given ideal, returning <tt>true</tt> if both inequalities hold and <tt>false</tt> otherwise. <tt>multBounds</tt> prints the bounds and the multiplicity (called the degree), and it calls <a href="_mult__Upper__Bound.html" title="determine whether an ideal satisfies the upper bound of the multiplicity conjecture">multUpperBound</a> and <a href="_mult__Lower__Bound.html" title="determine whether an ideal satisfies the lower bound of the multiplicity conjecture">multLowerBound</a>.</p>
<div>This conjecture was proven in 2008 work of Eisenbud-Schreyer and Boij-Soderberg.</div>
<table class="examples"><tr><td><pre>i1 : S=ZZ/32003[a..c];</pre>
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<tr><td><pre>i2 : multBounds ideal(a^4,b^4,c^4)
lower bound = 64 degree = 64 upper bound = 64
o2 = true</pre>
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<tr><td><pre>i3 : multBounds ideal(a^3,b^4,c^5,a*b^3,b*c^2,a^2*c^3)
35 140
lower bound = -- degree = 27 upper bound = ---
2 3
o3 = true</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>Note that <tt>multBounds</tt> makes no attempt to check to see whether <tt>R/I</tt> is Cohen-Macaulay.</div>
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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__Upper__Bound.html" title="determine whether an ideal satisfies the upper bound of the multiplicity conjecture">multUpperBound</a> -- determine whether an ideal satisfies 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>
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<div class="waystouse"><h2>Ways to use <tt>multBounds</tt> :</h2>
<ul><li>multBounds(Ideal)</li>
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