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<head><title>AnnFs(RingElement) -- the annihilating ideal of f^s</title>
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<div><h1>AnnFs(RingElement) -- the annihilating ideal of f^s</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>AnnFs f</tt></div>
</dd></dl>
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<li><span>Function: <a href="___Ann__Fs.html" title="the annihilating ideal of f^s">AnnFs</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a polynomial in a Weyl algebra <em>A<sub>n</sub></em> (should contain no differential variables)</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>, an ideal of <em>A<sub>n</sub>[s]</em></span></li>
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<div class="single"><h2>Description</h2>
<div>The annihilator ideal is needed to compute a D-module representation of the localization of <em>k[x<sub>1</sub>,...,x<sub>n</sub>]</em> at <em>f</em>.<table class="examples"><tr><td><pre>i1 : R = QQ[x_1..x_4, z, d_1..d_4, Dz, WeylAlgebra => toList(1..4)/(i -> x_i => d_i) | {z=>Dz}]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : f = x_1 + x_2 * z + x_3 * z^2 + x_4 * z^3
3 2
o2 = x z + x z + x z + x
4 3 2 1
o2 : R</pre>
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<tr><td><pre>i3 : AnnFs f
2 2
o3 = ideal (d - d d , d d - d d , z*d - d , d - d d , z*d - d , x d +
3 2 4 2 3 1 4 3 4 2 1 3 2 3 2 2
------------------------------------------------------------------------
2x d + 3x d - z*Dz, z*d - d , x d + 2x d + 3x d - Dz, x d - x d
3 3 4 4 1 2 2 1 3 2 4 3 1 1 3 3
------------------------------------------------------------------------
2
- 2x d + z*Dz - s, 3x z*d - z Dz + x d + 2x d )
4 4 4 4 2 3 3 4
o3 : Ideal of QQ[x , x , x , x , z, d , d , d , d , Dz, s]
1 2 3 4 1 2 3 4</pre>
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<div class="single"><h2>Caveat</h2>
<div>The ring of <tt>f</tt> should not have any parameters, i.e., it should be a pure Weyl algebra. Also this ring should not be a homogeneous Weyl algebra.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Ann__I__Fs_lp__Ideal_cm__Ring__Element_rp.html" title="the annihilating ideal of f^s for an arbitrary D-module">AnnIFs</a> -- the annihilating ideal of f^s for an arbitrary D-module</span></li>
<li><span><a href="../../Macaulay2Doc/html/___Weyl__Algebra.html" title="name for an optional argument">WeylAlgebra</a> -- name for an optional argument</span></li>
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