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<head><title>multUpperHF -- test a sufficient condition for the upper bound of the multiplicity conjecture</title>
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<div><h1>multUpperHF -- test a sufficient condition for 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=multUpperHF(R,hilb)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, the polynomial ring in which one wants to work</span></li>
<li><span><tt>hilb</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a Hilbert function for <tt>R</tt> modulo a homogeneous ideal</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 the sufficient condition holds and <tt>false</tt> otherwise</span></li>
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<div class="single"><h2>Description</h2>
<div><p>This function uses <a href="_cancel__All.html" title="make all potentially possible cancellations in the graded free resolution of an ideal">cancelAll</a> to test a sufficient condition for the upper bound of the Herzog-Huneke-Srinivasan conjecture to hold for all quotients of polynomial rings with Hilbert function <tt>hilb</tt>. The conjecture, proven in 2008 work of Eisenbud-Schreyer and Boij-Soderberg, asserts that if <tt>R</tt> is a polynomial ring, and <tt>I</tt> is a homogeneous ideal in <tt>R</tt>, then</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 largest shift at step <tt>i</tt> of the minimal graded free resolution of <tt>R/I</tt>, <tt>c</tt> is the codimension of <tt>I</tt>, and <tt>e(R/I)</tt> is the multiplicity of <tt>R/I</tt>.</p>
<p><tt>multUpperHF</tt> forms the lex ideal corresponding to the Hilbert function the user inputs. Then it calls <a href="_cancel__All.html" title="make all potentially possible cancellations in the graded free resolution of an ideal">cancelAll</a> to make all potentially possible cancellations in the resolution of the lex ideal (as described in the <a href="_cancel__All.html" title="make all potentially possible cancellations in the graded free resolution of an ideal">cancelAll</a> documentation). <a href="_cancel__All.html" title="make all potentially possible cancellations in the graded free resolution of an ideal">cancelAll</a> prints the diagram with the cancellations. In addition, <tt>multUpperHF</tt> compares the multiplicity associated to the given Hilbert function to the conjectured upper bound from the Betti diagram with the cancellations. If the conjectured inequality holds, the function returns <tt>true</tt>, and the conjectured upper bound is true for all modules <tt>R/I</tt> with the given Hilbert function. If the inequality fails, then the function returns <tt>false</tt>, and the sufficient condition does not hold. (However, this of course does not mean that the conjecture is false for the given Hilbert function; it may be that the Betti diagram with the cancellations cannot exist.) Note that since <tt>hilb</tt> is a finite list, all modules with that Hilbert function are Artinian and hence Cohen-Macaulay.</p>
<div>See C. Francisco, New approaches to bounding the multiplicity of an ideal, <em>J. Algebra</em> <b>299</b> (2006), no. 1, 309-328.</div>
<table class="examples"><tr><td><pre>i1 : S=ZZ/32003[a..c];</pre>
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<tr><td><pre>i2 : betti res lexIdeal(S,{1,3,4,2,1}) --just to see the resolution of the lex ideal
0 1 2 3
o2 = total: 1 7 10 4
0: 1 . . .
1: . 2 1 .
2: . 3 5 2
3: . 1 2 1
4: . 1 2 1
o2 : BettiTally</pre>
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<tr><td><pre>i3 : multUpperHF(S,{1,3,4,2,1})
total: 1 4 5 2
0: 1 . . .
1: . 2 . .
2: . 2 4 1
3: . . . .
4: . . 1 1
degree = 11 upper bound = 21
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__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>
<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>multUpperHF</tt> :</h2>
<ul><li>multUpperHF(PolynomialRing,List)</li>
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