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<head><title>reductionNumber -- reduction number of one ideal with respect to another</title>
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<div><h1>reductionNumber -- reduction number of one ideal with respect to another</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>k = reductionNumber(I,J)</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></span></li>
<li><span><tt>J</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the reduction number of <i>I</i> (defined below)</span></li>
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<div class="single"><h2>Description</h2>
<div><p>reductionNumber takes a pair of ideals <i>I,J</i>, homogeneous or inhomogeneous (in the latter case the ideal is to be regarded as an ideal in the localization of the polynomial ring at the origin.). The ideal <i>J</i> must be a reduction of <i>I</i> (that is, <i>J⊂I</i> and <i>I</i> is integrally dependent on <i>J</i>. This condition is checked by the function <tt>isReduction</tt> (missing documentation<!-- tag: isReduction -->). It returns the smallest integer <i>k</i> such that <i>JI<sup>k</sup> = I<sup>k+1</sup></i>.</p>
<div>See the book Huneke, Craig; Swanson, Irena: Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge, 2006. for further information.</div>
<table class="examples"><tr><td><pre>i1 : setRandomSeed()</pre>
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<tr><td><pre>i2 : kk = ZZ/101;</pre>
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<tr><td><pre>i3 : S = kk[a..c];</pre>
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<tr><td><pre>i4 : m = ideal vars S;
o4 : Ideal of S</pre>
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<tr><td><pre>i5 : i = (ideal"a,b")*m+ideal"c3"
2 2 3
o5 = ideal (a , a*b, a*c, a*b, b , b*c, c )
o5 : Ideal of S</pre>
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<tr><td><pre>i6 : analyticSpread i
o6 = 3</pre>
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<tr><td><pre>i7 : j=minimalReduction i
3 2 2 3 2
o7 = ideal (50c + 42a - 41a*b - 15b + 39a*c - 22b*c, 2c + 45a + a*b +
------------------------------------------------------------------------
2 3 2 2
19b - 39a*c - 38b*c, - 32c - 4a - 42a*b - 32b - 16a*c + 31b*c)
o7 : Ideal of S</pre>
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<tr><td><pre>i8 : reductionNumber (i,j)
o8 = 1</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>It is possible for the routine to not finish in reasonable time, due to the probabilistic nature of the routine. What happens is that the routine <a href="_minimal__Reduction.html" title="minimal reduction of an ideal">minimalReduction</a> occasionally, but rarely, returns an ideal which is not a minimal reduction. In this case, the routine goes into an infinite loop. This will be addressed in a future version of the package. In the meantime, simply interrupt the routine, and restart the computation.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_analytic__Spread.html" title="compute the analytic spread of a module or ideal">analyticSpread</a> -- compute the analytic spread of a module or ideal</span></li>
<li><span><a href="_minimal__Reduction.html" title="minimal reduction of an ideal">minimalReduction</a> -- minimal reduction of an ideal</span></li>
<li><span><a href="_which__Gm.html" title="largest Gm satisfied by an ideal">whichGm</a> -- largest Gm satisfied by an ideal</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>reductionNumber</tt> :</h2>
<ul><li>reductionNumber(Ideal,Ideal)</li>
</ul>
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