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<head><title>integralClosure(Ideal,ZZ) -- integral closure of an ideal in an affine domain</title>
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<div><h1>integralClosure(Ideal,ZZ) -- integral closure of an ideal in an affine domain</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>integralClosure J</tt><br/><tt>integralClosure(J, d)</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_integral__Closure.html" title="integral closure of an ideal or a domain">integralClosure</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>J</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
<li><span><tt>d</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, optional, default value 1</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the integral closure of <i>I<sup>d</sup></i></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><a href="_integral__Closure_lp..._cm_sp__Keep_sp_eq_gt_sp..._rp.html">Keep => ...</a>, -- list ring generators which should not be simplified away</span></li>
<li><span><a href="_integral__Closure_lp..._cm_sp__Limit_sp_eq_gt_sp..._rp.html">Limit => ...</a>, -- do a partial integral closure</span></li>
<li><span><a href="_integral__Closure_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- control the algorithm used</span></li>
<li><span><a href="_integral__Closure_lp..._cm_sp__Variable_sp_eq_gt_sp..._rp.html">Variable => ...</a>, -- set the base letter for the indexed variables introduced while computing the integral closure</span></li>
<li><span><a href="_integral__Closure_lp..._cm_sp__Verbosity_sp_eq_gt_sp..._rp.html">Verbosity => ...</a>, -- display a certain amount of detail about the computation</span></li>
</ul>
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</li>
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<div class="single"><h2>Description</h2>
<div><div>The method used is described in Vasconcelos’ book, <em>Computational methods in commutative algebra and algebraic geometry</em>, Springer, section 6.6. Basically, one first computes the Rees Algebra of the ideal, and then one reads off the integral closure of any of the powers of the ideal, using linear algebra.</div>
<table class="examples"><tr><td><pre>i1 : S = ZZ/32003[a,b,c];</pre>
</td></tr>
<tr><td><pre>i2 : F = a^2*b^2*c+a^3+b^3+c^3
2 2 3 3 3
o2 = a b c + a + b + c
o2 : S</pre>
</td></tr>
<tr><td><pre>i3 : J = ideal jacobian ideal F
2 2 2 2 2 2 2
o3 = ideal (2a*b c + 3a , 2a b*c + 3b , a b + 3c )
o3 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i4 : time integralClosure J
-- used 1.48577 seconds
2 2 2 2 2 2 2
o4 = ideal (b c - 16000a*c, a c - 16000b*c, a*b c - 16000a , a b*c -
------------------------------------------------------------------------
2 3 2 2 2 5
16000b , a c - 16000a*b, a b + 3c , a b + 15997a*c)
o4 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i5 : time integralClosure(J, Strategy=>{RadicalCodim1})
-- used 0.697894 seconds
2 2 2 2 2 2 2
o5 = ideal (b c - 16000a*c, a c - 16000b*c, a*b c - 16000a , a b*c -
------------------------------------------------------------------------
2 3 2 2 2 5
16000b , a c - 16000a*b, a b + 3c , a b + 15997a*c)
o5 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i6 : integralClosure(J,2)
5 2 4 3 3 2 3 2 4 4 3
o6 = ideal (b c - 2b c - 16000a*b - 3a*c , a b c - 2a c - 16000b - 3b*c ,
------------------------------------------------------------------------
3 2 5 4 3 5 2 3 3 4
a b c - b c - 16000a + 16000a*b , a c - a b c - 16000a b + 16000b ,
------------------------------------------------------------------------
6 2 2 2 3 3 4 2 2 4 3 3
a*b*c - 16000a b c - 12000a c - 12000b c - 8003c , a b c - 16000a c
------------------------------------------------------------------------
3 3 2 2 3 3 3 2 4 2 2
- 16000b c + 8003a*b*c , a b c - 16000a b*c - 16000b c + 8003a*b c,
------------------------------------------------------------------------
3 2 3 4 2 3 2 2 4 3 2 2 2
a b c - 16000a c - 16000a*b c + 8003a b*c, a b*c + 3a b c +
------------------------------------------------------------------------
3 2 4 2 3 2 4 3 3 2 4 4
8003b c, a b c + 3a b c + 8003a , a b c - 16000a b*c - 16000a*b c +
------------------------------------------------------------------------
2 2 4 2 2 2 3 4 5 2 3 2 3 6 2
8003a b , a b c + 3a b c + 8003b , a b*c + 3a b c + 8003a*b , a c +
------------------------------------------------------------------------
4 2 2 3 4 4 2 2 3 2 2 4 3
3a b*c + 8003a b , a b c - 16000a b + 3a*b c - 15997a c , a b c -
------------------------------------------------------------------------
2 4 2 3 2 2 5 2 3 3 3 3
16000a b + 3a b*c - 15997b c , a b c - 16000a b + 3a c -
------------------------------------------------------------------------
2 4 4 2 2 2 4 7 2 2 2 3 3
15997a*b*c , a b + 6a b c + 9c , a b + 6a b c + 15997a c - 15997b c +
------------------------------------------------------------------------
4 8 4 2 3 2 5 2 7 3
9c , a b*c + 15988a c + 15997a*b c + 9a*c + 15988a b*c, a b +
------------------------------------------------------------------------
5 4 3 3 10 2 4 2 5
15988b c + 8021a - 27a*b + 15988a*c , a b + 27a b + 15988a*b +
------------------------------------------------------------------------
2 3 2 2
27a*b c - 7940a c )
o6 : Ideal of S</pre>
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<div class="single"><h2>Caveat</h2>
<div><div>It is usally much faster to use <tt>integralClosure(J,d)</tt> rather than <tt>integralClosure(J<sup>d</sup>)</tt></div>
</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_integral__Closure_lp__Ring_rp.html" title="compute the integral closure (normalization) of an affine domain">integralClosure(Ring)</a> -- compute the integral closure (normalization) of an affine domain</span></li>
<li><span><a href="../../ReesAlgebra/html/_rees__Algebra.html" title="compute the defining ideal of the Rees Algebra">reesAlgebra</a> -- compute the defining ideal of the Rees Algebra</span></li>
</ul>
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