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<head><title>mRegularity -- compute Castelnuovo-Mumford regularity</title>
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<div><h1>mRegularity -- compute Castelnuovo-Mumford regularity</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> mRegularity I</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a homogeneous <a href="../../Macaulay2Doc/html/___Ideal.html" title="the class of all ideals">Ideal</a></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the Castelnuovo-Mumford regularity of the given ideal, if it is homogeneous, and -1 otherwise</span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>CM => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, a parameter that should be set to <a href="../../Macaulay2Doc/html/_true.html" title="">true</a> if the ideal is known to be Cohen-Macaulay</span></span></li>
<li><span><tt>MonCurve => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, parameter that should be set to true if I is the ideal of a monomial curve</span></span></li>
<li><span><tt>Verbose => ...</tt> (missing documentation<!-- tag: mRegularity(..., Verbose => ...) -->), </span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><p>This package is based on two articles by Bermejo and Gimenez: <tt>Saturation and Castelnuovo-mumford Regularity</tt>, Journal of Algebra 303/2006 and <tt>Computing the Castelnuovo-Mumford Regularity of some subschemes of P^n using quotients of monomial ideals</tt>, Journal of Pure and Applied Algebra 164/2001.</p>
<p>computing the regularity of the defining ideal of the second Veronesean of P3</p>
<table class="examples"><tr><td><pre>i1 : R=QQ[a,b,c,d,x_0..x_9,MonomialOrder => Eliminate 4]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : i=ideal( x_0-a*b,x_1-a*c,x_2-a*d,x_3-b*c,x_4-b*d,x_5-c*d,x_6-a^2,x_7-b^2,x_8-c^2,x_9-d^2)
o2 = ideal (- a*b + x , - a*c + x , - a*d + x , - b*c + x , - b*d + x , - c*d
0 1 2 3 4
------------------------------------------------------------------------
2 2 2 2
+ x , - a + x , - b + x , - c + x , - d + x )
5 6 7 8 9
o2 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i3 : j=selectInSubring(1, gens gb i)
o3 = | x_5^2-x_8x_9 x_4x_5-x_3x_9 x_3x_5-x_4x_8 x_2x_5-x_1x_9 x_1x_5-x_2x_8
------------------------------------------------------------------------
x_4^2-x_7x_9 x_3x_4-x_5x_7 x_2x_4-x_0x_9 x_1x_4-x_0x_5 x_0x_4-x_2x_7
------------------------------------------------------------------------
x_3^2-x_7x_8 x_2x_3-x_0x_5 x_1x_3-x_0x_8 x_0x_3-x_1x_7 x_2^2-x_6x_9
------------------------------------------------------------------------
x_1x_2-x_5x_6 x_0x_2-x_4x_6 x_1^2-x_6x_8 x_0x_1-x_3x_6 x_0^2-x_6x_7 |
1 20
o3 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i4 : I=ideal flatten entries j -- this is the ideal of the Veronesean,
2 2
o4 = ideal (x - x x , x x - x x , x x - x x , x x - x x , x x - x x , x
5 8 9 4 5 3 9 3 5 4 8 2 5 1 9 1 5 2 8 4
------------------------------------------------------------------------
2
- x x , x x - x x , x x - x x , x x - x x , x x - x x , x - x x ,
7 9 3 4 5 7 2 4 0 9 1 4 0 5 0 4 2 7 3 7 8
------------------------------------------------------------------------
2
x x - x x , x x - x x , x x - x x , x - x x , x x - x x , x x -
2 3 0 5 1 3 0 8 0 3 1 7 2 6 9 1 2 5 6 0 2
------------------------------------------------------------------------
2 2
x x , x - x x , x x - x x , x - x x )
4 6 1 6 8 0 1 3 6 0 6 7
o4 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i5 : mRegularity I
o5 = 3</pre>
</td></tr>
</table>
<p>This is an example where mRegularity is faster than regularity. Regularity takes approximately 190 seconds.</p>
<table class="examples"><tr><td><pre>i6 : R = QQ[x_0..x_5]
o6 = R
o6 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i7 : I1 = ideal (x_0^2*x_1+x_0*x_1*x_2-x_0*x_4^2,-x_0*x_2^2+x_0^2*x_5,x_0^2*x_2-x_0*x_1*x_4,x_0^3-x_2^3+x_0*x_1*x_3,x_0^3+x_0^2*x_1-x_1*x_2^2-x_0*x_2*x_5,x_0^3+x_2^3-x_0*x_5^2)
2 2 2 2 2 3 3
o7 = ideal (x x + x x x - x x , - x x + x x , x x - x x x , x - x +
0 1 0 1 2 0 4 0 2 0 5 0 2 0 1 4 0 2
------------------------------------------------------------------------
3 2 2 3 3 2
x x x , x + x x - x x - x x x , x + x - x x )
0 1 3 0 0 1 1 2 0 2 5 0 2 0 5
o7 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i8 : benchmark "mRegularity I1"
o8 = .4389332
o8 : RR (of precision 27)</pre>
</td></tr>
</table>
<p>This is an example where regularity is faster than mRegularity.</p>
<table class="examples"><tr><td><pre>i9 : R = QQ[x_0..x_5]
o9 = R
o9 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i10 : I2 = ideal ( x_0^2+x_5^2, x_0^2+x_0*x_3+x_4^2, x_0^2+x_0*x_5+x_2*x_5, x_0^2-x_0*x_3-x_3*x_5, x_0^2-x_3*x_4, x_0*x_3);
o10 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i11 : benchmark " mRegularity I2"
o11 = .030190221
o11 : RR (of precision 27)</pre>
</td></tr>
<tr><td><pre>i12 : time regularity I2
-- used 0.002999 seconds
o12 = 4</pre>
</td></tr>
</table>
<p>This symbol is provided by the package Regularity.</p>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="../../Macaulay2Doc/html/_regularity.html" title="compute the Castelnuovo-Mumford regularity">regularity</a> -- compute the Castelnuovo-Mumford regularity</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>mRegularity</tt> :</h2>
<ul><li>mRegularity(Ideal)</li>
</ul>
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