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<head><title>WeylClosure -- Weyl closure of an ideal</title>
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<div><h1>WeylClosure -- Weyl closure of an ideal</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>WeylClosure I, WeylClosure(I,f)</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 left ideal of the Weyl Algebra</span></li>
<li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a polynomial</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the Weyl closure (w.r.t. f) of I</span></li>
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<div class="single"><h2>Description</h2>
<div>Let R = K(x<sub>1</sub>..x<sub>n</sub>)<d<sub>1</sub>..d<sub>n</sub>> denote the ring of differential operators with rational function coefficients. The Weyl closure of an ideal I in D is the intersection of the extended ideal RI with D. It consists of all operators which vanish on the common holomorphic solutions of I and is thus analogous to the radical operation on a commutative ideal.<p></p>
The partial Weyl closure of I with respect to a polynomial f is the intersection of the extended ideal D[f<sup>-1</sup>] I with D.<p></p>
The Weyl closure is computed by localizing D/I with respect to a polynomial f vanishing on the singular locus, and computing the kernel of the map D --> D/I --> (D/I)[f^{-1}].<table class="examples"><tr><td><pre>i1 : W = QQ[x,Dx, WeylAlgebra => {x=>Dx}]
o1 = W
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I = ideal(x*Dx-2)
o2 = ideal(x*Dx - 2)
o2 : Ideal of W</pre>
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<tr><td><pre>i3 : WeylClosure I
3 2
o3 = ideal (x*Dx - 2, x*Dx - 2, Dx , x*Dx - Dx)
o3 : Ideal of W</pre>
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<div class="single"><h2>Caveat</h2>
<div>The ideal I should be finite holonomic rank, which can be tested manually by holonomicRank.The Weyl closure of non-finite rank ideals or arbitrary submodules has not been implemented.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Dlocalize.html" title="localization of a D-module">Dlocalize</a> -- localization of a D-module</span></li>
<li><span><a href="_sing__Locus.html" title="singular locus of a D-module">singLocus</a> -- singular locus of a D-module</span></li>
<li><span><a href="_holonomic__Rank.html" title="rank of a D-module">holonomicRank</a> -- rank of a D-module</span></li>
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<div class="waystouse"><h2>Ways to use <tt>WeylClosure</tt> :</h2>
<ul><li>WeylClosure(Ideal)</li>
<li>WeylClosure(Ideal,RingElement)</li>
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